Very doubtful that a Quebecois regional party would emerge with any but a very late annexation PoD and the US electoral system in place. Why mainstream Quebecois voters should willingly consign themselves to political insignificance in Congress and Electoral College alike by voting for a powerless regionalist-minority party ?
About the OP question, and assuming an ARW PoD since I find it the most plausible one for a successful US-Canada union, I assume that Canadian electorate would relatively quickly align themselves with the developing US two-party system.
At the ARW, Canada was a mostly agrarian society dominated by the Church and the signeurs with some substantial commercial elites in Montreal and southern Quebec. IMO, once Canadians and the 13 colonies gets accustomed to their federal marriage, this should make Canada a mostly Democratic state with strong ties with the Southern agrarian elites (once they get accustomed to each other, Canadian seigneurs and Southern planters shall soon discover that they have many many interests and viewpoints in common). However, Canada also has a significant borgeusie urban element, which is only going to expand and grow in size and influence as integration with America propels economic and social development of the region. Moreover, American influence most likely shifts the tug of war between the urban elites and the Church for control of Quebec in favor of the former. Besides occasionally flaring commercial rivalries with NE and NY, these elites are natural allies of their American Northern counterparts, and the Federalists and later Whig and Republican parties. As they grow in economic and social influence, Canada shall gradually shift from being strongly pro-South and Democrat to being increasingly pro-North and pro-Federalist/Whig/Republican. Also the Canadian Catholic Church is likely to swing towards anti-slavery, albeit less radically than NE Protestants.
Therefore, Quebec shall most likely evolve during early 19th century from being a mostly pro-South swing state, to a true swing state, then a pro-North swing state, then a strongly pro-North state, however less radical than NE or NY. Eventually, it shall come to be a French-Catholic variant of the Mid West, and build its identity accordingly.
The fact that Canada shall reasonably quickly find a place in the American socio-political landscape shall put a powerful brake to regional separatism (they shall become the French-Catholic "cousins" of the South first, the more conservative ones of NE/NY second, eventually the half-siblings of the Mid West up to modern times) and ease their national integration (the presence of Lousiana and immigration shall have a similar effect as well). Therefore in the long term, Quebec separatism shifts more and more to the fringe as they shift to become closer to and more nationally integrated with the North. Since the Gilded Age, the "Old Canada" (Ontario-Quebec) North shall politically (and besides the Franco-Catholic elements, culturally) little distinguishable from the rest of the MidWest. Say, Quebec shall remain as culturally distinct as Texas. As it concerns the NorthWestern Canadian states, most likely they shall develop political and cultural patterns rather akin to the American north-western states.