THE MUSEUMS of the Sunbelt owe a debt to Washington's. Throughout the southern states, from Florida to Texas and on to California, important art museums -- in New Orleans, Atlanta, Los Angeles and Birmingham, in Tucson, San Antonio, Fort Worth and West Palm Beach -- are now being directed by skilled museum specialists who earned their reputations here.

The exodus continues. Two more Washington museum men will soon be heading south.

Peter C. Marzio, the tough-minded director of the Corcoran Gallery of Art, is leaving town for Houston. So is Walter Hopps, this city's foremost expert in contemporary art. Marzio will run the biggest art museum in the state of Texas, Houston's Museum of Fine Arts. Hopps will head the newest, the Menil Collection, whose $20 million home is now being designed.

The art museum business, like so many others, is thriving in the Sunbelt. Dallas and Los Angeles, Austin, Houston and Atlanta are getting new museums. The audiences are there, hungering for history. The money is there, too.

The late J. Paul Getty, who might have left his art -- and his $750 million -- in England, where he lived, chose Malibu instead. Wealthy Norton Simon, who owns art as good as Getty's, might have placed it anywhere, but selected Pasadena.

Texas has the Oilers, the Astros, the Spurs, the Mavericks, the Rangers, the Rockets -- and the Cowboys. It is just as rich in art. Architects as prominent as Mies van der Rohe, Louis Kahn and Philip Johnson have built museums there. Now, at vast expense, their admirable buildings are being filled with art. "Fort Worth," says Hopps, "is a city less prepossessing than Cincinnati -- but it's built four good museums." In 1980, one of them, Kahn's Kimball, paid $3.9 million for a landscape by Ce'zanne; last year the same museum paid a reported $6 million for a portrait by Velazquez; last month it spent a reported $4 million for "Nude Combing Her Hair," a 1906 Picasso bought from Norton Simon. No wonder art professionals are looking toward the South.

Once upon a time, the typical director of a provincial art museum was popularly supposed to be an import from Manhattan's Metropolitan Museum of Art. That is the case no longer. Today he is more likely to have been recruited here.

The skills the work requires -- how to raise funds, how to fire, how to hang an exhibition, supervise a staff, deal with the government, work within a budget, flatter a collector, soft-soap a trustee -- are skills not taught at Harvard. They are best learned on the job. And many jobs are here. "On an urban per capita basis," says Hopps, "there are more museum professionals employed in this city than in any other."

They learn from one another, they argue and they gossip, they scan each other's catalogues and gauge each other's shows. Unlike most art historians, who teach in universities, those employed in Washington must work to please the public. That shows up in their catalogues and in their installations. No scholar hard at work in Boston or Los Angeles, Chicago or Detroit, sees as many well-displayed, first-rate works of art.

For reasons of high protocol, most major exhibits from abroad move from capital to capital, and tend to open here. The American tax system, which promotes the migration of privately held objects into public hands, also favors Washington. Collectors who lived elsewhere -- among them Chester Dale, Lessing Rosenwald, Joseph Hirshhorn, and now Arthur Sackler, whose Oriental art is destined for the Mall -- perhaps felt that by giving to the capital they were giving to all Americans at once.

This city has been blessed, as well, with extraordinary teachers. The late Joshua C. Taylor of the National Museum of American Art, who may have been the best of them, taught Marzio, Hopps and many other young directors. "If Josh had lived another 15 years," says Marzio, "half of the directors of American museums would be people he had trained." J. Carter Brown, the director of the National Gallery of Art and a managerial master, has been another skillful guide.

Most museums nowadays are involved with the government in one way or another. Search committees know that. They recognize as well that a certain freshness still pervades Washington's museums. A number of the finest -- the National Museum of American Art, the National Portrait Gallery, the Hirshhorn Museum and Sculpture Garden, the National Gallery's East Building -- like most museums in the Sunbelt are still relatively new.

Ten museums in the Sunbelt are now guided by directors who learned their business here:

* The Norton Gallery of Art, West Palm Beach, Fla. The Norton's collection is particularly strong in modern French and 20th century American paintings. Though not large, it is choice. The 70,000-square-foot Art Deco gallery, which opened to the public in 1941, has an annual budget of $980,000 and a staff of 28. It may be the finest small museum in the Southeast. Its director, Richard A. Madigan, 44, who was assistant director of the Corcoran from 1963 to 1967, acknowledges the surprising number of former Washingtonians now running art museums in the South and the Southwest. "Our number is legion," he says.

* The High Museum of Art, Atlanta. The High, the largest museum in the Southeast, is busily expanding. Its new $20 million, six-level, white-tiled, 130,000-square-foot building is now under construction. Designed by Richard Meier (and partially funded by a gift from Coca-Cola's Robert Woodruff), it will open to the public in October, 1983. The High's director, Gudmund Vigtel, 57, spent nine years at the Corcoran, the last two as assistant director. He became the High's director in 1963.

* The Birmingham Museum of Art. Alabama's largest general art museum, the Birmingham has an annual operating budget of $1.4 million, and a staff of 35. It mounts some 25 shows each year. Its 108,000-square-foot building opened to the public in 1959. Its director, Richard Murray, 40, received much of his training, as did Marzio and Hopps, from the late Joshua Taylor. Murray studied with Taylor at the University of Chicago, and then followed him to Washington, to the Smithsonian Institution's National Collection of Fine Arts (now the National Museum of American Art). Murray spent nine years at that museum before moving to Alabama in September, 1979. "Washington, unlike New York, say, is nicely distant from the art market," says Murray. "Museum exhibits in Washington tend to rely more on brain-power than on money-power. One reason standards are so high there is that Washington curators know they have to live up to the expectations of their peers. If your colleagues pan you, it's tough."

* The New Orleans Museum of Art. Founded in 1910, the New Orleans museum is the largest art museum between Houston and Atlanta. It owns some 21,000 objects and has an annual budget of $2 million. Its director, E. John Bullard, 38, joined the National Gallery in 1968. He worked there as a museum curator in American art, then as Carter Brown's assistant, and finally as curator of special projects before moving to New Orleans to take charge of the museum there in 1973.

* The Houston Museum of Fine Arts. The biggest in Texas, it has an annual operating budget of $7.1 million. Mies van der Rohe designed its 136,000-square-foot plant, which was built in stages and finally completed in 1974. Peter Marzio, who will become its director on Oct. 1, studied at Juniata College in Huntingdon, Pa., before going to Chicago, where Taylor and Daniel J. Boorstin, now the Librarian of Congress, jointly supervised his doctoral dissertation. Marzio joined the Smithsonian's National Museum of History and Technology in 1969 and worked there, first as curator of prints, later as the chairman of the department of cultural history, until July, 1978, when he was appointed director of the Corcoran.

"I was in many ways an innocent when I came to the Corcoran," he says. "I don't think I'd ever met a trustee. I thought the development officer was in charge of the building. I hadn't hired, hadn't fired, I knew nothing about budgets. A certain admirable 'eggheadism' prevailed at the University of Chicago. We were taught to think clearly and to write well, but not to please the average guy. I learned that at the Smithsonian. But Smithsonian curators don't have to worry about money. They're given a budget and told to do the best they can. What I know about fund-raising and budgets and supervising staff, I picked up at the Corcoran. I learned on the job." Under Marzio's regime, the Corcoran raised sufficient funds to operate in the black, installed a $2 million air-conditioning system ($1.65 million of that amount already is in hand), and increased attendance fivefold. Rather than risk a threatened budget deficit for the next fiscal year, the Corcoran last month cut its staff by 14 per cent. "When I got the job in Houston," says Marzio, "the board chairman down there told the local press that I'd been selected for my 'solid management.' "

* The Menil Collection. Its 70,000-square-foot, $20 million building, now being designed by Italy's Renzo Piano, co-designer of the Centre Pompidou in Paris, will house the remarkable collection formed by the late John de Menil and his widow, Dominique. They bought important modern paintings by Mondrian, Matisse, Picasso and Magritte, newer pictures by such Americans as Barnett Newman, Frank Stella and Andy Warhol, as well as fine examples of African and Oceanic art. They also commissioned Houston's Rothko Chapel, and, as a memorial to Dr. Martin Luther King, Jr., placed Barnett Newman's "Broken Obelisk" (which once stood outside the Corcoran) on the lawn outside.

Despite generous gifts to other institutions (they gave one large Jackson Pollock to Marzio's museum, and bought another, "The Deep," for the Pompidou in Paris), the Menil collection remains one of the best in Texas. Walter Hopps, who has been named the museum's first director, came to Washington from California in 1967. He served here as director of the Washington Gallery of Modern Art, as director of the Corcoran, and later, under Taylor, as curator of contemporary art at the National Collection. Hopps is now organizing "The Automobile and Culture," a $1 million exposition which will be mounted by the new Museum of Contemporary Art in Los Angeles, whose new building is now being designed. His show will open in that city in 1984, ten days before the start of the Olympic Games.

* The Amon Carter Museum, Fort Worth. Founded in accordance with the will of Amon G. Carter of Fort Worth, the museum was known as the Amon Carter Museum of Western Art when its building, designed by Philip Johnson, opened to the public in 1961. Its collection has since broadened. Its director, Jan Muhlert, 39, spent seven years with Taylor here at the National Collection of Fine Arts. "In Washington," she says, "you weren't allowed to be too narrow. You weren't expected to work in your special area only. There were so many artists there, and so many art museums, that you were able to involve yourself with the entire arts community.

* The McNay Art Institute, San Antonio. Long the largest in that city, it houses the collection formed by Marion Kugler McNay, who died in 1950, and it occupies the Spanish-Mediterranen mansion where she lived. It owns some 1,500 objects and has an annual operating budget of $600,000. John Palmer Leeper, 61, the McNay's director, worked here for the Corcoran -- first as keeper of the W.A. Clark Collection and later as assistant director -- in 1948 and 1949. He worked in Pasadena from 1950 to 1953, before joining the McNay.

* The University of Arizona Museum of Art, Tucson. "In my opinion," says its director, Peter Bermingham, 44, "we have the best permanent collection in the state of Arizona." The Tucson museum has 17,000-square-feet of exhibition space and an annual operating budget of $300,000. Bermingham studied American art at the University of Maryland under William Gerdts before receiving his doctorate from Michigan. He, too, worked with Joshua Taylor here. He spent six years at the National Collection as curator of education before moving to Arizona in 1978.

* The Los Angeles County Museum of Art. The 212,000-square-foot musuem is the biggest museum in the West, and has an annual operating budget of nearly $13 million. Los Angeles County opened in 1965, and few museums anywhere have expanded with such speed. A 30,000-square-foot addition to its Ahmanson Wing is now under construction, and its four-story, 75,000-square-foot, $20 million Atlantic Richfield Gallery of Modern Art is now being designed. Its director, Earl A. "Rusty" Powell III, 38, was recruited for the job from the National Gallery of Art. Powell, who received his doctorate from Harvard in 1974, came to Washington two years later. He spent four years at the Gallery, his last as executive curator, working closely with Carter Brown and supervising major loan exhibits, before moving to Los Angeles in March, 1980.