The resignation of Francis D. Carter as director of the D.C. Public Defender Service follows months of disagreement with the agency's board of trustees over the hiring of minorities and the issue of Carter's responsiveness to the board.

Sources on the board said several trustees had been pushing for Carter's resignation. "Apparently the friction became more substantial and came to a point where it's fair to say that Frank didn't have sup port from enough of the board," said one trustee who asked not to be identified.

Asked to comment on why he resigned, Carter said, "It was a personal choice and I made the choice for my own future."

Carter, 38, who had been head of the agency since 1979, announced last month that he would leave the post to enter private practice as soon as a replacement could be found.

Sources said Carter had been under pressure to reach a goal whereby 60 percent of agency lawyers would be blacks or members of other minorities. About a third of the lawyers are black now. Some persons familiar with the hiring issue said the board has been divided over the question of meeting the affirmative action goal versus hiring white lawyers who might be more experienced or appear more qualified.

"I am very concerned about the agendas that people on the board have," said lawyer Lawrence H. Schwartz, who left the board last year. "There are agendas there other than quality."

The defender service is an independent city agency with 61 lawyers who each year represent more than 1,000 indigent criminal defendants charged with serious crimes.

A replacement for Carter is being sought by the 10-member board, which consists of lawyers and others appointed by Mayor Marion Barry and the chief judges of the local and federal trial and appeals courts.

"The board feels that Frank did a tremendous job," said lawyer Vincent H. Cohen, chairman of the board. "He wasn't fired. And clearly he wasn't fired for failing to hire black lawyers."

According to several sources, the board established a goal in 1983 of achieving 60 percent minority representation in the agency in five years. Carter said that the agency had five minority lawyers when he took office and that there are now 20.

The new policy would have required that upwards of 70 percent of all newly hired lawyers be minority-group members.

Several sources said Carter ran into trouble with the board because he continued to hire whites. But those sources added that in numerous other matters, Carter frequently went his own way and irked board members by refusing to confront them directly over policy differences.

"If he didn't like it, he just wouldn't do it," said one board member. "Frank's style has always gotten him into trouble, not his performance."

Sources said the disagreements sometimes stemmed from relatively minor issues, such as a suggestion from board members that Carter enforce a dress code among the defense lawyers.

"There had been complaints that some of the lawyers were visiting clients in blue jeans and sweaters," said one board member. "Frank felt it was none of the board's business, that the clients should be happy just to have a lawyer. To me, it was a minor flap, but it's an example of how the board can get upset."

One board member said Carter was "grilled" by some of the trustees over his decision in 1983 to use a Supreme Court ruling that raised questions about the city's home rule authority as a basis for challenging the government's prosecution of a sex offender.

This member of the board said trustees who were sympathetic to Mayor Barry were pressing the city's position that the District's charter was unaffected by the high court's ruling.