The National Governors' Association yesterday endorsed freezes on defense spending and Social Security cost-of-living allowances and a tax increase, if necessary, as part of a plan to reduce the federal deficit.

But a majority of the governors voted against the controversial Social Security proposal, which President Reagan has opposed; it survived because a two-thirds vote was necessary to kill it.

The overwhelmingly Democratic organization for the first time backed passage of a balanced-budget amendment to the Constitution, which Reagan favors. In a day marked by partisan wrangling, the group then fell one vote short of calling on Reagan to submit a balanced budget to Congress.

The governors' group also put itself on record as favoring tax simplification. The vote came despite objections from some governors that simplification would mean elimination of the deduction for state and local taxes paid.

Yesterday's closing session was politically the most combative of the three-day winter conference here.

On Sunday, the association's executive committee endorsed the deficit reduction recommendation in a unanimous voice vote and soundly defeated, 7 to 2, a proposal to exempt Social Security cost-of-living adjustments (COLAs) from the plan.

But on Monday, the governors met with Reagan at the White House. Citing his landslide victory in last year's election, Reagan told them he planned to stand firm on his campaign promises not to cut Social Security or his defense buildup and to raise taxes only as a last resort.

Republican governors, appealing for unity and saying that the deficit remains one of the nation's main economic problems, insisted yesterday that Reagan had shown a willingness to respond to bipartisan initiatives from the governors. Democrats disagreed.

"We asked him twice whether he would agree to some compromise involving Social Security and defense , and he said, 'No,' and he said it as specifically as he could say it," New York Gov. Mario M. Cuomo told his colleagues. "What you're suggesting now is either that the president spoke cynically, which I reject -- I think he spoke sincerely -- or that we go through some kind of political charade for a purpose I don't understand.

"The president of the United States instructed us that he would not change his mind on this issue. He won; my side lost. I'm with the president . . . . I'm not going to ask anybody to join me in a statement against Social Security when there is no likelihood -- no likelihood at all -- that it's going to occur."

A majority of governors went on to vote against the Social Security proposal but fell three votes short of the two-thirds required to remove that provision from the plan. Florida Gov. Robert Graham (D), whose state is a haven for retirees on Social Security, led the fight against the freeze.

The governors then voted, 26 to 14, for a resolution calling on the president to submit a balanced budget. Kansas Gov. John Carlin, chairman of the association, was the only Democrat to vote against that proposal, which fell one vote short of the two-thirds necessary for adoption.

"I would hope that we don't get in this business of gamesmanship on the deficit . But if we're in it, let's all get in it together," said Gov. Richard W. Riley (D) of South Carolina, who introduced the proposal on behalf of the Democratic Governors' Association.

North Carolina Gov. James G. Martin (R) opposed the measure, describing it as an effort to "tweak the president's nose." And Pennsylvania Gov. Richard L. Thornburgh (R) said it would be unrealistic to expect a balanced budget "overnight."

On Sunday, in a gesture to Reagan and Republican governors, the executive committee had voted to include in its package support for the balanced budget amendment plus line-item veto authority for the president, which Reagan also wants. But by yesterday, this cooperative mood among the Democrats had soured.