BDM International Inc., a Northern Virginia professional services company whose stock began trading yesterday on the American Stock Exchange, said profits last year rose about 15 percent on a modest increase in revenues that reflects a slowdown in some consulting work.

President Earle C. Williams, in New York for opening trading ceremonies on the Amex floor, said management figures indicated profits for 1981 of $3.15 million to $3.23 million (a range of $1.42 to $1.46 a share) compared with $2.76 million ($1.50) the previous year.

Per-share earnings are down even though overall profitability is up because BDM sold an initial offering of common stock to the public last year, increasing to 2.13 million shares from 2.02 million the number of shares outstanding.

Revenues last year were in a range of $87-$89 million compared with $83 million in 1980, Williams said in an interview.

BDM provides diversified technological and analytical services to government agencies, business and foreign nations from more than 20 offices in the United States and overseas.

The McLean firm's stock opened at $15.50 on the Amex and closed its first day up 25 cents a share at $15.75. But the real gain for the day was 75 cents, since BDM had closed in the over-the-counter market at $15 bid on Tuesday afternoon, its last day as an OTC issue.

First American Bank of Virginia reported a 31 percent jump in profits last year compared with 1980. The Financial General Bankshares unit in Northern Virginia said net income was $12 million ($6.74) compared with $9.1 million ($5.13). Assets of the McLean bank rose 8 percent in 1981 to $960 million on Dec. 31.

In the fourth quarter alone, First American profits rose to $2.79 million ($1.57) from $2.38 million ($1.33). In comparison to the sharp year-to-year gain, the relatively flat final quarter "may suggest a much less impressive year in 1982," said President Milton Drewer Jr.

For the year, net interest income rose $6.8 million in 1981, with much of the growth coming from an unusual sector--real estate. Drewer said his bank made $67 million of construction loans last year and more than $36 million of home mortgages to 352 buyers. Deposits rose 6 percent to $821 million and total loans expanded by 13 percent to $553 million.

Bowl America Inc. of Springfield, an operator of bowling centers, said profits rose in the six months and second quarter ended Dec. 27. Net income for the recent three months was $541,404 (40 cents) compared with $491,620 (36 cents) a year ago as sales rose to $4.67 million from $4.36 million.

Earnings for the first six months of the company's fiscal year were $510,643 (38 cents) vs. $405,807 (30 cents) as sales rose to $7.8 million from $7.2 million. Six-month earnings are lower than in the recent quarter because bowling firms traditionally have less business in the summer months covered by the first quarter, which often produces a net loss.

Government Employees Financial Corp., a consumer finance subsidiary of Geico Corp., posted a net loss of $484,000 in 1981 compared with a loss the previous year of $1.45 million. The loss in the fourth quarter alone was $434,000 compared with a loss of $240,000, as consumer loan firms generally have suffered under the weight of record interest costs. Revenues for the year fell to $36.2 million from $39.6 million, as Geico management sought to reduce the volume of loans outstanding.