Washington Post Co. profits nearly tripled in the first quarter as the result of sharp increases in advertising and circulation at The Washington Post and the sale of two unprofitable operations, Inside Sports magazine and the Trenton Times newspaper.

The Post company's net income climbed to $5.1 million (36 cents a share) for the 13 weeks ended April 4 from $1.8 million (13 cents) in the same period a year ago. Earnings for the year will be "substantially higher" than last year's, but "the rate of increase for the year will not be as great" as the first quarter's gain.

Revenues for the quarter rose 10 percent to $183.1 million from $165.8 million, while operating revenues climbed to $7.8 million from $4.7 million.

A spokesman said the jump in earnings "was due to strong performances by the company's newspaper and broadcast divisions" and the sale of the two losing ventures late last year. A gain of 8 cents a share resulted from the sale in March of the company's 24 percent share in Publishers Phototype Inc.

Newspaper division revenues increased 21 percent "reflecting an increase in advertising linage, rate increases and continuing large circulation gains" for The Post since the demise of The Washington Star last year. Magazine revenues "were up only slightly," and advertising in U.S. editions of Newsweek was down 11 percent.

In other earnings reports yesterday, NS&T Bankshares Inc. posted a 25 percent gain in earnings but Dominion Bankshares said its profits fell 22 1/2 percent because of losses on low-interest mortgages.

Dominion Bankshares, based in Roanoke, had net income of $4.2 million (49 cents a share) compared with $5.5 million (64 cents) for the similar period last year. Assets totaled $2.8 billion, loans amounted to $1.6 billion and deposits were $2.2 billion, all gains from the first quarter of 1981.

Dominion's "operating performance continues to reflect pressures resulting from high interest rates, a continuing shift to higher cost deposits and a relatively large portfolio of long-term, low-yielding assets, especially mortgage loans," the company said.

D.C.-based NS&T's profits grew 25.3 percent during the first quarter, to $1.5 million ($1.07) from $1.2 million (85 cents) for the same period last year. Loans climbed to $408 million, deposits increased to $481 million, and total assets are up to $578 million.