Just one day after President Carter unofficially kicked off his campaign here in this first-in-the-nation primary state, the New Hampshire Senate today returned the kick by adopting a balanced-budget resolution vehemently opposed by Carter.

The vote makes New Hampshire the 30th state to approve a resolution calling for a constitutional convention to require a balanced federal budget. Only four more states are needed to force the convention, although the validity of some of the resolutions is under question.

President Carter yesterday reaffirmed his commitment to a balanced federal budget. However, he and a recently formed coalition headed by Massachusetts Lt. Gov. Thomas P. O'Neill III, son of the U.S. House speaker, have attacked the method of calling a convention, the first in over two centures, for its potential danger to the Constitution.

The 16-to-7 vote and a favorable vote by the state's House last week were also seen as a test of political strength between Carter and his likely challenger in the primary only 10 months away, California Gov. Edmund G. (Jerry) Brown Jr., who has lobbied heavily for the resolution.

"It's a referendum on the president's economic policy and the verdict is guilty," said Mark Bodi, a Democratic state representative and Brown supporter. "It is definitely a slap in the face for the president, shedding a real positive light on Brown and enhancing his potential for running."

Brown, who telephoned most of the Democratic state senators from California this week while his local operatives worked on the Republicans, is expected to announce his candidacy in June or July, according to his chief of staff Gray Davis, who added, "this vote is unquestionably a win for Jerry Brown."

"It is a self-anointed victory; Jerry Brown is a self proclaimed ayatollah," countered House Minority Leader Chris Spirou, a staunch Carter supporter. "He is flexible for any religion: Proposition 13 or a balanced budget. If retroactive abortion became politically popular he'd be in favor for that, too."

Added Republican Senate President Robert B. Monier; "Brown is trying to claim credit for something that would have passed anway. He hasn't had anything to do with it; if you ask the people of New Hampshire if they wanted a balanced federal budget, 75 percent would say, 'Hell yes and fast."

Spirou tried to stop the drive in the Senate this week by suggesting to two senators who strongly supported Carter in the last presidential contest here that they submit an amendment to gut the resolution.

Sen. Louis E. Bergeron, Democratic Senate vice president, said, "Spirou told me I would be guaranteeing the reflection of Carter and Gov. Hugh Gallen if I did it; that it would make me a hero and I'd get anything I wanted."

He refused as did Sen. Robert Fenelly, Carter's New Hampshire coordinator in 1976. "Brown has won a major victory her today," said Fennelly, who no longer supports Carter. "We haven't seent the last of him in New Hampshire."