Stressing their common heritage and the need unite under one banner, Hispanic Democrats from across the nation met here over the weekend to test their political muscle and adopt strategies aimed at flexing it.

The national convention of Hispanic American Democrats (HAD) brought together for the first time leading activists from the Chicano, Cuban and Puerto Rican communities in a forum designed to eliminate differences and cement alliances both for the 1980 elections and for the long term.

"This is a very historic day because this is the day we as a group are putting our act together," said Rep. Robert Garcia (D-N.Y.), in a speech Friday opening the convention for more than 300 people from 12 states and Puerto Rico.

"We've always given our votes to the Democratic Party. Now we have to make that party responsive to us," said David Lizarraga, chairman of HAD and a Democratic Party leader in southern California, stressing one of the convention's central themes. "We have to claim what's ours."

Although Hispanic Americans are the fastest growing minority in the United States -- official estimates place the number at 12 million, but unofficially there are thought to be millions more -- individual communities have tended to work at cross-purposes rather than hand-in-hand with other Hispanics, convention participants said.

"The burning issue to Chicanos has been undocumented workers, while the issue for Cubans has been political prisoners, and for Puerto Ricans, statehood," explained Henry Casso, president of a Chicano career guidance firm in Albuquerque.

As a result, "turf problems" have developed, Lizarraga said. The feeling has been that now, "New York belongs to the Puerto Ricans, Florida belongs to the Cubans, the Southwest belongs to the Chicanos," with each group unwilling to come to the aid of others, he said.

But now the various Hispanic groups are realizing, as Lizarranga put it, "the power we have is so limited, we can't afford to divide it."

The theme of the three-day meeting was not only politics, but developing political sophistication. HAD agreed not to endorse a candidate for the Democratic presidential nomination for the time being, preferring to have Jimmy Carter, Massachusetts Sen. Edward M. Kennedy and California Gov. Edmund G. (Jerry) Brown Jr. court the group's favor.

Representatives of the three candidates met convention delegates today to make presentations.

Another convention goal was to build a base from which Hispanic delegates to next summer's Democratic National Convention in New York could be elected.

Lizarraga, hopes to have as many as 300 delegates at the convention to push HAD programs through the national platform committee.

In nonpolitical actions at today's closing session, HAD approved resolutions calling for economic sanctions such as boycotts against employers who do not hire Hispanics, a presidential commission on Hispanic education and representation on the Federal Communications Commission.