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Residential neighborhoods in this troubled province tend to be segregated along religious lines, and the visitor can often tell, just by looking around, which side he is on.

One clue is the Union Jack, which flies only in Protestant neighborhoods. Another is whether the graffiti that can be seen everywhere are directed against the Pope or against Queen Elizabeth II.

Yet another clue is the children. There are usually many more of them on the streets of a Roman Catholic neighborhood, because the Catholics tend to have bigger families. On that demographic fact of life here rest political implications of enormous significance.

The Catholics are still in the minority in Northern Ireland, but their proportion of the population is growing. Many Protestants regard the steady increase in the Roman Catholic population as a demographic time bomb. 'Better Begin to Adjust'

''Inevitably, there will be a Catholic majority in the north within my lifetime,'' said a Protestant college student here, adding that ''people my age better begin to adjust to the idea.''

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Successive British leaders have pledged that this province will remain British for as long as its majority wishes. But if most of those who could vote were Catholic, it would presumably choose union with the Irish republic.

For various reasons relating to the sectarian violence, statistics on religion and population are inexact in Northern Ireland. But it is thought that in 1978, for the first time, more children were born to Catholic parents than to Protestants. There were 26,239 births that year, and the Roman Catholic Church recorded 13,286 baptisms.

According to Paul Compton, a professor of geography at Queen's University here and a student of the population trends, the birth rate among Catholics is 21 per thousand, as against 14 per thousand for Protestants. Because the Catholic population has become progressively younger, its death rate is considerably lower than that of Protestants - 9 per thousand as against 13 per thousand. Exact Figures Uncertain

Mr. Compton estimates that Catholics made up 38 percent of the population in 1979. But no one knows for sure, and the results of the 1981 census appear likely to shed little light on the matter.

Not only is the question about religion voluntary, with many people refusing to answer it, but the whole census, like so much in Northern Ireland, is charged with sectarian politics as well.

Some militant Catholic nationalists who refuse to recognize the sovereignty of the British here boycotted last year's census for political reasons, as they boycott elections conducted by what they regard as an illegal government.

In the census, a door-to-door canvasser in a Roman Catholic neighborhood of Londonderry was stopped on the street and shot dead. That halted the census there and discouraged other census takers from being overly thorough in similar neighborhoods.

Another unknown factor is emigration from the province. Usually, more Catholics than Protestants have left. But ''the troubles,'' as the last 12 years are generally called, have produced dissatisfaction and fear in a previously complacent Protestant population. Change Now Seems Possible

Although the Protestants, the side that wants Northern Ireland to remain British, still constitute a majority, Prime Minister Margaret Thatcher tacitly acknowledged for the first time last November that it was possible that the majority here might change.

In a joint communique with the Prime Minister of Ireland, she reiterated the ''guarantee'' of British rule here for as long as the people wished it. Then the communique added that if the people of Northern Ireland ever voted in favor of changing the constitutional status of this province, ''the British Government would of course accept their decision and would support legislation in the British Parliament to give effect to it.''

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