In a South that was almost solidly for Jimmy Carter in 1976, there still is firm support for the President among the Democratic governors' Conference in Hilton Head, S.C.

It is, for course, a commentary on the first-term problems of the Carter Administration that the governors' approval of a Democratic president who recently served as a southern governor himself should be worth noting at all.

However, only a year ago, at the Southern Governors' Conference in San Antonio, the South's chief executives seemed to be right in step with the swing of public opinion that led to sharp decline in Carter's ratings.

"We didn't have to say a thing about the President at San Antonio," South Carolina's Republican Governor James B. Edwards said last week. "The Democrats did it for us."

At Hilton Head, it was a different story. To be sure, the President made gubernatorial approval easier with the dramatic announcement of the Camp David accords the night before the conference opened.

The governors warmly commended this achievement in a resolution passed at their first session. And at their last session, amid some misgivings, they even approved a motion by Tennessee's irrepressible Ray Blanton nominating Carter for the Noble Peace Prize.

However, Carter supporters at Hilton Head insisted in interviews that their backing of the President does not depend entirely on his role in the Mideast negotiations. Moreover, governors Dolph Briscoe of Texas and Edwin W. Edwards of Louisiana said their support is strong enough to survive their disappointment over the Administration's natural gas pricing proposals, which Briscoe and Edwards regard as nothing less than punitive to their states.

The governors' endorsements of Carter seemed to rest primarily on two judgments they have made about him: First, he is willing to take high political risks to cope with heretofore intractable problems in a "hostile" Washington environment, and second, he retains the ability to rally the voters to his side.

Speaking to the first point, Alabama's George C. Wallace sat on an eighth floor balcony overlooking the Atlantic Ocean at Hilton Head and ticked off Carter's fights for a Panama Canal Treaty, Civil Service reform, an energy bill and a Mideast settlement.

"He has gone into some situations where the chances for political success were low and the chances for political disaster were high," he said. "But he has gone ahead and not worried about the political outcome . . . I don't say that I necessarily approve of all he has done . . . but I know he dares to tackle problems that a politician might ought to leave alone."

Among the Southern governors, there is a scarecely concealed belief that the President's problems have been exaggerated by the Washington establishment, including the press.

"I think his problems have been overstated," West Virginia's John D. Rockefeller IV said. "Washington is a hostile environment for a president, especially in the first two years of his administration."

Rockfeller offered both the Camp David accords and the likelihood of a Carter victory in the form of a natural gas bill as evidence of solid Carter accomplishments.

North Carolina's James B. Hunt Jr. is even more scornful of press treatment of Carter. When it was suggested that the press naturally has focused on the President's performance in trying to filfill expectations he raised in his campaign, Hunt replied, "Well, I guess that's as good an excuse as any."

Hunt and Kentucky's Julian M. Carroll are among those who remain confident that Carter will rebound with the voters, despite the results of approval polls. "He never has lost ground in Kentucky," Carroll said. "Our most recent poll shows him with a 66 percent approval rating."

"When he came to my home town of Wilson," Hunt said, "the reception was tremendous. Among the grassroots, he's still their man. That's a fact that not everyone in Washington understands."