The Prince George's County Council yesterday approved a $551 million budget for next year and used a massive infusion of state aid to reduce homeowners' taxes by the largest amount in county history.

The budget, approved 10 to 1 by the council after a curtailed review process, would save the owner of a $65,000 home $108 in taxes next fall if his property was not reassessed this year. The county rate was reset at $2.68 per $100 of assessed valuation, 35 cents below the current tax.

In one week of work sessions, the Democratic council members made relatively few adjustments to the budget plan submitted last month by their frequent political adversary, Republican Executive Lawrence Hogan.

The tensions between Hogan and the council nevertheless showed no sign of slackening. Hogan yesterday quibbled with several budget changes the council had made, and the council used its afternoon session to reject a Hogan nominee to the Washington Suburban Sanitary Commission for the second straight time.

"The Council acted irresponsbily in both its review of the budget and the conclusions it reached," a Hogan press release declared late yesterday.

Hogan and other county officials once had predicted drastic cuts in government services as a result of the tax-limiting TRIM county charter amendment adopted by county voters in 1978.

However, the budget adopted yesterday will result in no budget cutbacks in county services or programs.

"It's clearly what the people wanted," said council chairman Parris Glendenning. "A tight budget, but maintaining essential people-serving, street-level services."

The council's budget will not eliminate any jobs presently filled, but will cut funding for hundreds of positions throughout the county government that are now vacant.

The budget also retains funding for the 14 schools Hogan had recommended closing.

In Montgomery County, by contrast, the council has approved a $662 million budget for next year, but county Executive Charles Gilchrist has recomended a tax increase of 14 cents. The concil has not yet formally set t the tax rate for next year.

Prince George's ability to cut its taxes while other jurisdictions are raising them is attributed not to TRIM, but, largely to a $15 million increase this year in the amount of aid the county receives from the state.

That windfall which came to the county largely in the form of increased funds for schools, police and transportation, is not likely to happen again in the near future, county officials acknowledged.

At least one council member also was openly concerned that the county's tight reign on its departmental budgets, though not resulting in noticeable cuts this year, eventually would damage citizens' services.

"I hope citizens understand that if they call a policeman and he doesn't show up, it may be because his car broke down," said veteran Councilmember Francis B. Francois. "And if we have a particularly bad winter, potholes will be there this time next year. We are postponing costs, not getting rid of them."

Hogan was displeased with the Council's actions for different reasons. "In seeking to pay lip service to reducing taxes, and at the same time knuckling under to the demands of special interests, [the Council] has achieved the worst of both worlds." Hogan's statement said. "Its tax reduction could have been considerably larger; its reallocation of resources was inappropriate, imprudent and short-sighted."

In order to give county residents a large tax cut, Hogan proposed across-the-board reductions. In most county agencies, including the elimination of more than 900 positions in county government that now are vacant, closing 14 public schools and scaling down several parks programs.

The council elected to keep the schools open and retain the parks programs, and did so by increasing Hogan's broad cuts in most agencies.

Most significantly, the council slashed $188,647 from the budget of the Department of Plannng and Economic Development, the source of Hogan's prized programs for promoting county business and development.

The overall growth in the budget over last year's figure is .6 percent, an amount both council members and Hogan say is mild considering the inflation rate.

Among the budget's major expenditures are $287 million for county schools, $41.5 million for police and fire protection: $3.9 million for the Corrections Department, and $11.6 million for the Health Department.

Among anticipated revenues are $145.4 million from the county property tax, $82.6 million from the state income tax, $152 million in state aid and $26.2 million in federal aid.

In its other major action yesterday, the council voted 8-to-2 to reject Hogan's nominee for the WSSC, the Rev. John F. Warren, of Hillcrest Heights, to replace Andrew Vislosky.