The board of directors of Geico Corp., the holding company that owns the Government Employees Insurance companies, voted yesterday to increase the quarterly dividend from 10 cents to 11 cents -- the third increase in 18 months.

Geico Chairman John J. Byrne announced the increase at an annual shareholders meeting at the company's headquarters in Chevy Chase, where he also pointed to signs of the company's relative health.

"Our surplus for the protection of policyholders has topped $250 million, higher than at any point in our 43-year history," Byrne said, pointing to a graph that showed a sharp climb from what he called "the rocky valley" of 1976.

It was in 1976 when Geico's fortunes were so uncertain that many people thought the firm's full name was "the financially troubled Geico Corp." -- a year when Washington's biggest insurer report major losses and angry shareholders turned out in force at the annual meeting to question the competency of management.

Part of the company's problems then stemmed from practice of writing policies worth more than the amount of money set aside to pay for claims.

Besides pointing to the growth in policyholders' surplus, Byrne called 1979 "a reasonably prosperous year" in other respects. Although net income declined in 1979 and for the first quarter of 1980, per share earnings have grown, Byrne said. The growth resulted in large part from the company's policy of buying back its common stock and reducing the number of shares outstanding.

"Our shares of common and common-share-equivalents went from 34.2 million at the end of 1978 to 25.1 million at year end 1979," Byrne said. Since then the number of outstanding shares has been reduced even further to 23.7 million.

Byrne also told shareholders that return on equity "has hovered around 40 percent for the past three years, a notable achievement that puts us in the top ranks of American business. The company expects the rate of return to decline, but Byrne said he expects Geico can continue to earn above 20 percent.

That kind of return is "absolutely necessary if we are to deserve our shareholders' capital in the decade ahead," he said.

"Prospects for the property casualty business are clouded by the economic uncertainities that surround us," Byrne said. He said he expected gains in productivity and continued attention to costs to carry Geico through the uncertain future.