Since the Colonial age the Quebecois have referred to themselves as "les habitants" (Fr. the inhabitants). When the NHL was first formed the Montreal Canadiens were intentionally a predominantly French Canadian team, whereas English speaking Montrealers had their own team, the Montreal Maroons. As a result, it was natural to call les Canadiens by the nickname of the people they represented. (Habs is the short form.)
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Having said that, the nickname might have been a result of an error. According to NHL.com, the first man to call the Canadians "Habs" was Tex Rickard, owner of the Madison Square Garden in 1924. Rickard apparently told a reporter that the H on the Canadians sweater was for Habitants. Not true the distinctive C-wrapped-around-the-H logo stands for "Club de Hockey Canadian".
The Montreal Canadian were found has the Club de Hockey Canadien de Montréal. At this time Canadian was the word to describe the French population in the province of Québec.