Fairfax County Executive J. Hamilton Lambert proposed a "cautious" $1.4 billion budget yesterday that would expand day care, police and fire protection and services for the elderly and mentally ill while leaving the county's real estate tax rate unchanged.

The fiscal 1987 budget represents a spending increase of 6.3 percent.

Although he termed the county's fiscal health "superb," Lambert set aside a special $7 million reserve to offset potential shortfalls from cuts in federal and state aid to the county stemming from the Gramm-Rudman-Hollings balanced budget act.

"Gramm-Rudman could have severe ramifications on the county," Lambert warned. Fairfax County receives about $33 million annually in federal grants, subsidies for mass transit and construction, housing programs and school food programs.

Some Fairfax politicians said yesterday they still hope to enact a modest tax cut this year, but others predicted that the budget-cutting mood in Congress would undermine any chance that the County Board of Supervisors might reduce county taxes significantly.

"I think the board would like to cut the tax rate," said Supervisor Martha V. Pennino, its Democratic vice chairman. "But Gramm-Rudman may mean that we can't do it."

Board Chairman John F. Herrity, a Republican, disagreed, saying, "I personally believe that we can cut taxes . This is a basic policy and we will continue it this year."

Several supervisors said the board is likely to cut personal property taxes -- the tax paid by county residents on automobiles -- and suggested they might be able to slash Lambert's proposed $7 million Gramm-Rudman reserve to make room for that cut.

Some board members also noted that county budget officials have consistently underestimated revenues. Lambert acknowledged recent projections showing fast economic growth in Fairfax that could produce $6.1 million in additional income for the county in fiscal 1987.

But Herrity and other supervisors said it was unlikely that any tax cut could offset increased taxes caused by higher real estate assessments countywide.

Even if the real estate tax rate of $1.39 per $100 of assessed value remains unchanged, a typical county homeowner will have to pay slightly higher taxes because of higher assessments. The tax bill for the owner of a residence assessed at $109,051, for example, would be $1,516 next year, an increase of $54.

The county has cut the tax rate in six of the last nine years, but increasing property values meant that homeowners watched their tax bills rise steadily. Last year's 6 cent decrease in the tax rate was the first real average reduction the county has had in recent times.

Lambert's proposed budget, which goes to the County Board for final action in late April, includes major service expansions in public safety, transportation and human services. Among the major new programs:

*$4 million to convert the Lincolnia School into a 54-unit housing project for the elderly.

*$286,000 for four new day care centers for school-age children, including one for the handicapped, and expansion of five existing centers.

*$1.9 million for 72 new positions in the police and fire departments.

*$3.6 million to replace two aging fire stations in Clifton and Chantilly.

*$4 million for unspecified county road construction projects.

*$2.7 million for the expansion of the Franconia government center.

The proposed budget also calls for additional positions and services in a new crisis care center, two mental health centers and a new program for patients with both emotional problems and drug dependency.

In all, the budget includes funds for 275 new county positions.

In addition, Lambert proposed spending $5 million to buy a fleet of buses to establish a county-run feeder bus system to serve the Vienna Metrorail station. He said the bus service probably would be similar to a system started at the Huntington Metrorail station last year.

Lambert's budget did not include funds for the proposed new Fairfax government center near Fair Oaks Mall, which the county hopes to build as a joint venture with a developer in return for a grant of land now owned by the county. Last year, Lambert included $14 million for road and sewer construction at the government center site, a proposal that was rejected by the board when it sought cutbacks to make room for a tax decrease.

Questioned about the absence of funds for the center, which would cost tens of millions of dollars to build, Lambert said he now expects a private developer will pay for roads and sewers at the site.

Also absent was any mention of Fairfax County's school teachers, who are demanding sharply higher salaries than those proposed by the County School Board. The teachers are expected to focus their lobbying efforts on the Board of Supervisors in coming months and some supervisors say that issue could dominate their deliberations on the budget.