Fairfax County supervisors will be asked Monday to appropriate $1.5 million for design of the long-stalled Dulles Toll Road, a commuter roadway that would be the Washington area's only toll road and the first major highway financed in part by the county.

County Executive J. Hamilton Lambert has urged the county board to get into the highway construction business by approving a design contract for the $54 million, four-lane road. It would parallel the little-used Dulles Access Road which the federal government owns and, despite frequent protests, has restricted largely to Dulles International Airport traffic.

The toll road, connecting with a new section of Interstate 66 inside the Capital Beltway, would permit nonstop travel between the Roosevelt Bridge and the Loudoun County line, relieving horrendous congestion around Tysons Corner and on Route 7.

Lambert has recommended that the board approve the contract without competitive bidding on an emergency basis to Dewberry & Davis, a Northern Virginia engineering firm. Lambert said the speed is necessary to take advantage of depressed construction prices and to allow the road to open by the fall of 1984.

The state legislature has approved plans to finance construction of the road with revenue bonds, but those plans have been delayed because of soaring interest rates, which have increased the projected automobile tolls from 50 cents to as much as 85 cents for a one-way trip. Top Fairfax officials lobbied intensively in Richmond for the past several months in an effort to get the project moving.

Reston commuters and business interests desperately want the 13-mile road, but the supervisors, worried about the political fallout, have been reluctant to discuss publicly a proposal to spend as much as $5 million in local funds on what would normally be a state project.

"I hate taking the highway department off the hook, but that's the only way we're going to get roads," Supervisor Martha V. Pennino, who represents Reston, said recently. "This isn't the greatest project in the state of Virginia from the highway department's view, that's the feeling I get."

Northern Virginians long have accused the state highway department of neglecting their needs, and the resentment has intensified as the department's funds have dwindled. Last year the county won the right to sell bonds to finance its own secondary road construction, and its contribution toward the toll road will mark a major step toward assuming a burden that had always fallen on the state.

Pennino, Supervisor Nancy K. Falck and top county officials traveled to Richmond almost every week during the recently-concluded General Assembly session, meeting with state Sen. Adelard L. Brault and Virginia highway officials in an effort to secure a combination of state and local money that would guarantee construction of the road. Board Chairman John F. Herrity once called the toll road "one of the most important issues if not the most important issue for Fairfax County facing the General Assembly this year."

Despite its high priority to the board, the supervisors rarely discussed in public the bill that Brault introduced and shepherded through the legislature allowing Fairfax to spend its own money on primary roads.

Herrity and Pennino say the toll road will spur economic development in the vast tracts of industrial-zoned land around the airport. The toll road has been opposed by homeowners along its path and by some friends of the concert area at Wolf Trap Farm Park, a few hundred yards from the proposed roadway. Proponents say sound barriers can be built that will protect the summer music arena.