Canadian Prime Minister Pierre Trudeau is going to Washington Monday to meet President Carter for the first time and to tell Americans that Canada can survive as a united country.

Trudeau and Carter are expected to hold four hours of discussions during the 2 1/2-day visit, and Trudeau will address Congress Tuesday, the Canadian prime minister ever to do so.

Officials in both countries say that the economic and cultural squabbles of recent years are largely past and that the two leaders will be sizing each other up to see how well they can get along.

They list a range of subjects to be discussed, from environmental problems along the U.S.-Canadian border to world arms limits - but they say that U.S. interest will focus on Trudeau's response to Canada's constitution crisis created by the election of a separatist government in Quebec Province.

Trudeau insists that he will not use his 20-minute speech to Congress to reply directly to Quebec Premier Rene Levesque, who went to New York last month to tell influental bankers and investors there why he wants to make the French-speaking province an independent republic.

The speech to Congress is seen here by many as potentially one of the most important of Trudeau's political career. It will be broadcast live on the country's public-owned national television network.

Levesque urged members of the Economic Club of New York to continue investing in Quebec. He said their investments would be secure, even if Quebec gains independence and the 110-year-old Canadian confederation collapses. There were outraged responses from Canadian federalists here over his comparison of his campaign for independence with the American Revolution.

Canadian authorities say the crisis over Quebec will be "of intense interest" in the United States.

"He can't reply directly to Levesque but he will want to tell Carter and the Congress of his expectations for Canada's future," a senior offical said. "There is bound to be a lot of interest in the national political integrity of this country."

State Department official say Americans will want to hear where Trudeau thinks confederation is going and what it means for the United States.

"Instinctively Americans are not separatists and will listen very attentively to what Trudeau says about the future of Canada," one said.

Trudeau is said to be worrying over how to present his case for national unity at home while speaking in a foreign country.

He can easily fuel further separatist sentiments in Quebec by coming down too hard on Levesque. He cannot understate the seriousness of the crisis, however, particularly following a decision this week by the Royal Bank of Canada, one of the country's largest chartered banks, to move part of its head office operation from the Quebec metropolis of Montreal to Toronto, the most important city of English Canada. The move includes the bank's investments, international money markets and corporate marketing and development sections.

The Trudeau visit is regarded here as the beginning of friendier Canadian American relations. For the first time, there is no list of so-called "irritants" to be dealt with.

"It is fair to say that several years ago there was some concern on both sides over the way the relationship was going," a U.S. official said. "But now attitudes have changed and there is a more favorable feeling."

U.S. officials say they have adjusted to Canada's gradual phase-out of oil exports and its increase in oil and gas prices to world market levels.

American fears over restrictions on foreign investment in Canada have subsided and the cultural feud that centered around the tax status of Time magazine has ended. Time killed its Canadian news section but continues to profit from domestic advertising in the U.S. edition sold here.

There has been a temporary truce in the dispute over U.S. border television stations selling advertising in Canadian markets. Canada has stopped randomly stripping commercials from programs beamed into the country by U.S. stations.

Both countries expect further progress this year on a five-year-old joint program to clean up the Great Lakes.

In addition, a major decision will be made this fall on the transportation of natural gas from Alaskan gas fields. The alternatives area trans-Alaska gas pipeline with liquefied natural gas shipped to U.S. destinations in tankers, or a pipeline through the Mackenzie River valley in the Northwest Territories.

There will be discussions of the proposed Garrison Dam project in North Dakota, which is seen by Canada as a potential pollutant of rivers in Manitoba Province.

The number of day-to-day contacts between top government officials in both countries increased dramatically with the arrival of Gerald Ford in 1974, and the expectation is that Trudeau's relations with Carter will be relaxed and friendly.