Geico Corp. customers across the country will be able to "call on Geico any day, any time," as the Chevy Chase insurance firm plans to introduce its widely advertised 24-hour-a-day telephone service nationwide, a company official said yesterday.

The announcement follows a heavily advertised trial of the system which began in October in the Washington area. Under the plan. Geico automobile and home insurance customers and potential customers can call a local number at any time to report claims and seek information on new or existing policies. Geico officials said their system will be the only one of its kind in the field.

Edward H. Utley, regional vice president for Geico's Macon, Ga.-based district operation, said in an interview yesterday that the new service would be operated toll-free and would ultimately link his office's southeastern location with the rest of the nation.

Utley said the trial has been successful here and that a customer survey indicates that "many people in the Washington area think it's a benefit, whether they use it or not."

The initial marketing of the service is expected to begin in several states with significant numbers of Geico policyholders, such as New York, Florida, Texas and California, Utley said.

Utley's comments came during an interview after the company's annual stockholders' meeting, which was held at Geico headquarters yesterday. The session was generally upbeat and, in fact, Geico Chairman John Byrne used the forum to point out that the company's fortunes have turned around dramatically since the mid-1970s, when Geico was in serious financial trouble.

The company "has earned the right to consider the last five years a closed chapter of the Geico story," Byrne said. Since that time, Geico has "tenaciously observed disciplined financial controls," rearranged its investment portfolio, improved the status of Geico's sister companies and reduced a backing outstanding claims, Byrne said.

The company's profits, however, dropped last year to $60.7 million from $74.2 million in 1979. But sales rose to $726.4 million from $705.8 million and the company's stock has shown a steady increase in book value. Property and casualty written premiums rose by 5 percent and investment income rose by 6 percent last year.

Byrne was asked about his last year's salary, which totaled $782,875, more than four times that of any other officer of the company. That figure combines a $150,000-a-year salary, which was raised to $225,000 for 1981, with a bonus of $600,000 and other considerations.

Byrne called his compensation "generous," and told a stockholder that his salary is "tightly tied" to the company's peformance.

For the first quarter of 1981, Geico profits rose slightly from $16.1 million (65 cents a share) to $16.4 million (75 cents), the company said in a report to stockholders. Total revenues for the first quarter rose to $184.5 million, up from $177.9 million for the 1980 first quarter.

The company yesterday also declared a quarterly cash dividend of 12 cents a share on its $1 per value common stock, payable june 30 to stockholders of record on june 11. The company also announced a quarterly cash dividend of 18 cents a share on its convertible preferred stock, payable July 1 to stockholders of record on June 11.