Chester C. Carter, a former State Department protocol official who later owned a liquor distributorship here, is expected to be named next week as the new executive director of the D.C. Lottery Board.

Carter would replace Douglass W. Gordon Jr., who, according to a source, would continue to work for the board in a diminished role, probably on lottery research.

The 61-year-old Carter, who was a deputy assistant secretary of state in the Johnson administration and later deputy chief of protocol, said final details of his hiring have yet to be worked out with the lottery board.

But he said, "It sounds like a challenge. I'm going to give it a shot."

Carter would take over the day-to-day administration of the city's only legally sanctioned gambling at a time when the board is deep in preparation for the expected summer start of a daily numbers game similar to the one now operating in Maryland.

In this type of game, players pick their own three-digit number and then wait to see if their number is drawn at the end of the day.

The lottery board's 1,100 ticket agents have sold more than 36 million $1 instant lottery tickets since last August during two games in which players scratch off a covering to find out if they have won a cash prize of as much as $20,000.

The lottery board hired a private company, Games Production Inc., to operate these games under a one-year contract, but in the coming months the board has to decide whether to assume direct operational control of the instant games.

Carter said he has played the instant lottery "religiously" but has yet to win more than $2 on any one ticket. He joked that in his new job, since he is forbidden from buying lottery tickets, the "worst thing is that I can't win anything."

He will, however, be paid $56,000 annually, the same salary that Gordon has been paid.

Based on the experience of states that operate daily numbers games, D.C. lottery officials expect that revenues will far exceed those generated by the instant games. As a result, lottery board members have said they felt they needed a stronger chief executive than they say Gordon has proved to be since he took office last February.

None of the board's current 28 full-time and temporary employes has had experience running a lottery, nor has Carter.

After leaving the State Department in the late 1960s, Carter worked as an executive in the international division of the Joseph E. Seagram & Sons distilling corporation and then owned the Capitol City Liquor Co. Inc., a liquor distributor, for 10 years before folding the company in 1980.

The source said the lottery board believes that experience should help smooth the start-up of the daily numbers operation since many of the first group of 300 computers that will be used to write the numbers tickets will be installed in D.C. liquor stores.

The source said that Gordon "has not been able to manage that office" and that "we were never able to get our work out."

The source said the lottery board's staff was supposed to have its supplemental budget for the current fiscal year and its 1984 budget prepared by late November, but did not finish the task until just before Christmas.

Gordon said earlier this week he expects to stay with the lottery board, but had not been told of any impending changes. He could not be reached for comment yesterday.