The Fairfax County School Board voted last night to ask state officials to remove Fort Belvoir from the county school system in a move designed to counter proposed federal budget cuts for the education of children of military families.

"It's time the federal government stopped trying to play Robin Hood with Fairfax County," said board vice chairman Mary E. Collier of the Dranesville district.

If the proposal is approved by various elements of local and state governments, Fort Belvoir would be set up as a separate school district. Federal and state governments would be responsible for picking up the entire bill for educating the 2,000 children whose families live on the Army base.

Congress now is considering Reagan budget proposals that would sharply limit the federal impact-aid program. The program is designed to subsidize the education of children of military families in school systems across the nation.

Fairfax school officials are concerned that they could lose much of an anticipated $4.4 million impact-aid allotment under pending congressional proposals.

The board voted 7 to 2 to support the staff recommendation, with members Carmin Caputo of the Centreville district and Gerald Fill of Mt. Vernon voting no.

Caputo said, "It would allow us to wash our hands of a sticky situation. Rather than taking the easy way out, we should try to work out an agreement with the federal government."

"We've talked enough," retorted member James W. Kitchin, of Mason District. He noted that the board had been haggling with various federal agencies for six months, demonstrating that "trying to negotiate is absolutely fruitless."

The proposal must be approved by the Fairfax County Board of Supervisors and the State Board of Education. The Virginia General Assembly has veto power over such a proposal.

Earlier, the Fairfax board threatened to charge tuition fees to the 2,000 children who live at Fort Belvoir unless Congress appropriated funds for at least half the cost of educating them. The Justice Department has filed suit against the proposal.