First Virginia Banks Inc. of Falls Church said yesterday its earnings remained strong in the second quarter, with profit at $16.8 million (79 cents a share), compared with $16.6 million (79 cents) for the same quarter last year.

The bank company attributed the stability to its consumer loan portfolio, which has continued to provide income at a time when many area financial institutions are being hurt by the real estate slowdown.

For the first six months, First Virginia reported a profit of $32.8 million ($1.54), down 1 percent from $33.2 million ($1.58) in the first half of 1989.

Chairman Robert H. Zalokar said weaker loan demand and a lower interest margin affected earnings growth, but he added, "the highly publicized loan problems of many banks, particularly in the real estate area, are not a major concern in First Virginia's banks."

The bank had assets of $5.2 billion as of June 30, up 7 percent from $4.8 billion held at the same point last year. First Virginia operates 305 branches in Virginia, Maryland and Tennessee.

Palmer National Bank reported yesterday that it earned $183,319 in the second quarter, a 3 percent decline over the same period a year ago. The company attributed the decrease to an addition to its loan loss reserves, the cushion that protects against possible loan defaults.

The privately held, District-based institution said it earned $432,018 in the first six months of the year, an 11.6 percent increase over the first half of 1989. Through its one-branch office downtown, Palmer provides private banking services to individuals, partnerships and corporations.