While the pilgrims are largely perceived as having moved to the US primarily for religious freedom (among many other social and economic factors) the people who were sent to New France were generally single males. For much of the early French colonial endeavor, the colonization of what is now Quebec was meant to be extractive and strategic rather than being a settler colony.

This saw a predominant focus on the fur trade, lumber exploitation and military posts rather than more sedentary settlements such as those of the pilgrims. So to answer your question, many moved to Quebec for money, some for a better standard of living, others for adventure, others still because they were sent.

While Quebec did eventually progress to a settler colony it happened relatively late which allowed for a relatively easy British conquest during the Seven Year's War. When offered a choice between Quebec and the Caribbean colonies of Martinique and Guadeloupe in the postwar settlement, France chose Martinique and Guadeloupe due to their importance as sugar colonies over Quebec. So Quebec did indeed not gain its independence, rather it changed hands in the global imperial race.

Hope this helps!

1. The French first colonized Canada in the 1600s. And they had a hard time getting people to move. As to the techniques they used, see the Filles Du Roi. France also encouraged the first settlers to breed with the First Nations, though it isn't very well known.
2. Quebec never gained it's independence from France. They abandoned us to be slaughtered by the invading english in the 1750s.

PS: The term is Québécois (Kay-bay-kwa), not Quebecian.