A limited partnership headed by New York developer Alan Stillman plans to build a $19 million office building at 1110 Vermont Ave. NW, current site of the Vermont Avenue parking garage and a small building formerly owned by savings and loan executive Thomas J. Owen and others. The $3.3 million property transaction was handled by Robert L. Cohen of Barnes, Morris and Pardoe Inc. Construction is expected to begin this month. To the north, on the site of the old Burlington Hotel, another large office building is scheduled to be started theis spring.

Hugh T. Peck Properties Inc., a real estate company with offices in Maryland, plans to open eitht offices in Northern Virginia this year. The first will be opened in Vienna in April. The Potomac office of the Peck firm recently handled the sale of the Bullis School property at 8501 Houston St., Silver Spring, to the Christian Services Corp. for $980,000. The organization, a missionary peace corps, will move from its headquarters on 16th Street downtown.

A course on renting and managing a home in Northern Virginia will be given on Sunday afternoons, March 11 and 18, at the Annandale campus of Northern Virginia Community College. The course, to be taught by tealtor William R. Bonner, is designed to introduce home owners and small investors to the management of residential property. For information, call 323-3168.