Virginia National Bankshares Inc., the state's second largest bank company, reported fourth-quarter income of $7.8 million ($1.03 a share), a 24 percent gain over the $6.2 million (85 cents) earned in what the bank said was a depressed final three months of 1981.

The $3.9 billion Norfolk company had fourth-quarter profits of $9.2 million before securities transactions ($1.22), a 48 percent increase over the $6.2 million (86 cents) in 1981.

Bank of Virginia, the third largest bank, said its income in the fourth quarter was $6.8 million ($1.17), a 28 percent increase over the $5.3 million (95 cents) it earned before securities transactions and $5.1 million (92 cents) after those transactions in the last three months of 1981.

C.A. Hutchins III, chairman and chief executive, said the sharp increase in the last three months of 1982 came because it compares with a 1981 quarter in which earnings were depressed.

For 1982 as a whole, Virginia National reported profits of $31.4 million ($4.23) compared with $27.6 million ($3.83) in 1981. Before securities transactions, Virginia National had $33.3 million ($4.48), a 14 percent increase over the $29.2 million ($4.06) the bank company earned in 1981.

Virginia National's board also authorized the company to buy back some of its shares from the open market to be used in the company's dividend reinvestment plan and its employe thrift plan.

For 1982, Bank of Virginia said its net income was $21.7 million ($3.77), compared with $15 million ($2.68). Before securities transactions, Bank of Virginia earned $24 million ($4.18), 46 percent higher than the $16.4 million ($2.93) it earned in 1981.