The state capitol and City Hall sit only four blocks apart in this flourishing mid-South city, and yet the views could hardly be more different.

Richard H. Fulton (D), 54, halfway through his second term as mayor of the consolidated metropolitan government of Nashville and Davidson County, looks up the hill at the capitol and sees indifference and neglect.

Lamar Alexander (R), 41, in the third year of his first term as governor of Tennessee, looks down at City Hall and sees unrealistic expectations and ineffectual representation.

Fulton and Alexander, as both men are eager to let you know, regard each other as allies more than political enemies. They say with some justification that they get along better than most mayors and governors.

But th tensin in their relationship is characteristic of a problem that is bound to grow in importance as the federal budget cuts recommended by President Reagan and approved by Congress filter down to the state and local levels.

That issue will be center-stage starting today at the National Governors Association annual meeting, to be held this year in Atlantic City.

Gov. Richard A. Snelling (R) of Vermont, the incoming chairman of the association, is pointing the organization toward a serious exploration of the differences that have emerged these past seven months between local and state goernments on the shift in size and shape of federal aid grants.

Fulton has been invited to Atlantic City in his role as the chairman of the policy committee of the U.S. Conference of Mayors. Alexander will be prominent as the liaison from the governors' executive committee to the White House.

The two men represent opposing views on the basic wisdom of Reagan's recasting of the federal aid system. This fiscal year the federal government is expected to give more than $90 billion in grants to state and local governments.

Such grants, for everything from Medicaid to highways, will make up about one-seventh of the federal budget, and provide more than one-fourth of all the money state and local governments have available to spend.

About one-third of Reagan's budget cuts for next fiscal year came in these grants programs, reducing aid to states and local governments about 15 percent below the level projected by Jimmy Carter's last budget. Almost everyone expects even deeper reductions in fical 1983 and 1984.

Fulton says the immediate cost is about $6.8 million in assistance. The bigger long-range problem as Reagan cuts back federal categorical aid and shifts to state-run block grants is that Nashville will have to turn increasingly to the state capitol, where "historically we have not done too well."

Alexander, a longtime proponent of block grants and the transfer of discretionary authority from Washington, says his administration has been more than responsive to the cities, and if the legislature has at times been negative, "The fault is in the urban delegations themselves."

Thus, the tension that is visible here is, if less embittered by rival political ambitions than in many other states, a laboratory study of the feuds likely to be increasingly visible as the Reagan program takes hold.

Nashville-Davidson County (the two governments were merged 18 years ago) has a population of 455,000 roughly 10 percent of Tennessee's population of 4.59 million. Metro Nashville contains all or a portion of the districts of 11 of the 99 state representatives and four of the 33 state senators.

Along with such other major cities as Memphis, Chattanooga and Knoxville, its delegation comprises part of an "urban bloc" with a theoretical 40 percent share of the legislature.

But this year Alexander's first-ever urban package was given short shrift, and almost everything on Fulton's wish list went unanswered.

Of the four major "local control" items the governor gave the legislature in january, only one, a penny increase in the state gas tax earmarked for city and county roads, got through. Proposals for a 1-cent local option gas tax, for an expansion of the local business tax an easing on the current $7.50 cap on local sales taxes were sidetracked or defeated.

Fulton says the $1.2 million Nashville will get from the extra penny of gas tax is the first such boost from the state in all his years as mayor.

"I think the governor has indicated his support of programs that would help Nashville," he concedes. "But, historically, Nashville has not received the consideration it should have received [from the legislature], and this year was no different."

Fulton's wish list included Alexander's three local-option items, plus a local-option 1 percent occupational privilege (payroll) tax that would have generated $50 million in revenues, a half-dozen other minor revenue measures and a $24.7 million bond issue for construction of a major bridge project. None got through.

There is a good deal of finger-pointing and fault-finding among all the participants. Fulton concedes that the city lobbyist, who also represented a Japanese auto maker building a Tennessee plant, might have been more diligent, but says Alexander did nothing effective when Republican legislators "ducked out" on the local-option tax package.

Alexander, in turn, points out that Democrats control both houses of the legislature and that Nashville legislators chair the committees or subcommittees where several of the local-option bills were pigeonholed.

The Nashville delegation, he notes, voted against the 1-cent increase in the state gas tax earmarked for local roads, the one item of help fulton acknowledges receiving from the state.

State Rep. James R. McKinney (D), chairman of the House Democratic caucus and dean of the Nashville delegation, one of the targets of Alexander's criticism, says the reason they voted against the gas tax was that they were unable to change its rural-oriented distribution formula.

And he throws the overall criticism back in Alexander's face: "He proposed some bills he knew would have no chance of passage, just for the sake of the rhetoric; but the one bill that would have been most important to the urban areas, the payroll-tax option he rejected."

Alexander says he came out against the payroll tax because it had no realistic prospects and because "It would add too much revenue to government at a time when we should be limiting the growth of government." Nashville, he notes, has one of the highest income levels in the state and one of the lowest property tax rates among the nation's major cities.

Behind all this debate, there is a fiscal-political reality that emerges from the talk of politicians, community leaders and knowledgeable journalists. Nashville, they say, is not a distressed city, and it could do more for itself; but boosting taxes is not popular at the local level either.

Conceding this, they say, there is also some merit to the mayor's complaint that Nashville and other cities face bleak prospects as they turn from Washington to the state capitol as the source of redistributed revenues and program funds.

One man who was willing to speak for attribution was Lewis Donelson, a Memphis attorney and Republican powerhouse who just returned to private practice after serving for the first 30 months of Alexander's term as the state commissioner of finance and administration.

Conelson brings a unique perspective as a self-described "city man," a former chairman of the Memphis City Council budget committee, but a GOP partisan and a key player in Alexander's dealings with the Democratic legislature.

His description of the Tennessee revenue-redistribution system:

"On the gas tax, the urban areas are getting raped horribly.The formula is so rural-oriented we've had some rural counties that never spent a cent on their own roads until we imposed a 20 percent matching requirement on them . . . . On school funds, they're distributed on a per-capita basis, which looks equal, but our minimum-support program has no adjustment for the cost-of-living difference between city and country or the complexity of urban education. So it's equal on paper and unequal in reality.

"On the other hand, the rural view is that the distribution of sales tax by county of origin favors the urban centers, because people drive in from the country to the big shopping centers. So if you take the cap off the local sales tax, as we proposed, rural people feel they are contributing to the Shelby County [Memphis] or Davidson county coffers. And the rural people have no interest in expanding business taxes because, frankly, they aren't collecting the ones on the books now."

Donelson gave this picture of the realities of fiscal politics in the Tennessee legislature:

"On paper, the urban counties could come close to a majority, but in fact the suburban Republicans vote like the rural Democrats. In my part of the state, and I suspect others, there's a little of the racist thing around.It used to be overt, but now it's covert, but they don't want to do all that much for the blacks in Memphis.

"Nashville is the most liberal city in Tennessee and the most unionized. What they're afraid of, and they have right to be afraid, is the general attitude in the legislature, that when the federal government gets out of an area, the state won't take it up either. That's where you get the main difference between the mayor and the governor.

"Medicaid is a good example. The feds are cutting the funds, and we've been trying to cut the state costs ever since we got in, with the help of the legislature. Now the cities say, and probably correctly, that this forces the costs back on them for indigent patients.

"We're cutting our Medicaid hospital limit from 21 days to 14 days, and that is going to cause trouble in Memphis and Nashville, because their hospitals serve the rural patients. Any time a rural hospital gets a burn case, whup, they put him in an ambulance and zap him down to Nashville or Memphis. They're say they don't have the facilities for treatment, but really, they don't want the cost."

These underlying realities are likely to be forced to the surface as the 25 percent cutback in major federal aid programs reaches the states and cities this fall and the first steps in the conversion from categorical to block grants take place.

Alexander says he is determined to show that many of the fears expressed by the mayors are unfounded. He gave The Washington Post figures showing that for the last five fiscal years the four urban counties, with just over 40 percent of the state populations, had received between 47 and 48.1 percent of the state-distributed revenues.

Last year two-thirds of the state's capital expenditures program went to the four counties, he said, and next year that will rise to 88 percent because of a $20 million redevelopment project targeted at downtown Memphis and its high black unemployment.

Alexander also ordered a fast start on planning for block grants, establishing an interdepartmental task force under Donelson that started work in February on tentative reprogramming decisions on education, health and social services funds.

There were informal discussions with local advisory groups, but no formal consultations with mayors on how the funds will be allocated. Donelson says, "I think when they see what we're proposing, the cities will feel they have been reasonably dealth with, but they won't like the fact that we have not replaced the missing federal dollars."

Alexander flatly rules that out for now. "Over the next few years," he said, "if President Reagan is successful in cutting a family's taxes $1,000, I have no objection to raising state taxes $200 or letting the locals raise that much for local purposes.But the legislature and I both are opposed to any state add-ons now. We don't believe it reduces the size of government to change taxes from one level to another."

McKinney says the Democratic legislators feel the same way. "We may try to pick up some of the cutbacks in exceptionally needy programs, but mostly, we'll let the cuts go through so people can feel the impact of what Reagan is doing."

Down at City Hall, Fulton says the measurable first-year impact on the $350 million budget will probably require the layoff of less than 100 employees. What he cannot calculate is what the Medicaid cutbacks at state and federal levels will do to his hospital costs, how much welfare may rise with the end of Comprehensive Education and Training Act (CETA) jobs, whether the money will be there for long-planned airport modernization and mass-transit projects.

And then there are all the special considerations he has been able to gain as a well-connected Democratic mayor, an alumnus of the House Ways and Means Committee, with friends all over the federal bureaucracy.

"Our riverfront park was the No. 1 priority for the Atlanta region on the urban parks program. We've gotten about $250,000 and we hoped for $5 million or $6 million eventually. I doubt that's going to be a priority in the eyes of the state legislature.

"Our Meharry Medical College and Hubbard Hospital train half the black physicians in America. We've always been able to go to Washington for help. Wilbert Cohen, when he was HEW secretary, took a personal interest, and came out here often to see what we were doing. It would be a real quesiton if we get that same kind of help from the state."

That fear, at least, is justified. Two weeks ago, on the final day of its session, the legislature failed to override an Alexander veto of a $150,000 appropriation for sickle-cell anemia research at Meharry.

Alexander said, "Meharry gets so much more from the state than any other hospital, I just couldn't justify any more. We have to set priorities."