It would be interesting how annexing Canada would affect US politics as a whole. A US State of Quebec would be solidly majority francophone (and I strongly doubt they would let Quebec go independent, if I remember right War Plan Red called for all Canadian provinces to be turned into US states), so while there would be an effort to assimilate them (in the sense of encouraging English-speaking and, on an unofficial level, discouraging Catholicism), they would see quite a bit of success in resisting these efforts. Canada has already tried to do the same to them, so they certainly wouldn't be strangers to anglophone federal governments trying to impose their language on them. There would be a solid attempt at cultural and linguistic assimilation during the period of military occupation, but as soon as that ends I predict that pro-francophone parties would form and get elected to the state government. Court challenges would abound, but the state government, like many state governments in OTL, would be fiercely protective of its autonomy. While in OTL all states that have designated an official language have designated English as such, I'm fairly sure there are no constitutional barriers to designating another language as official, like French. So Quebec would probably be free to do so.
Quebec in the modern day would probably look fairly similar to the way it does as a province in OTL; a culturally and linguistically unique state where French is the dominant language, and larger cities (where immigrants concentrate) are bilingual to some extent. Independence referendums might happen every now and then like in OTL, though given that the US has had the example of the Civil War as to what happens to secessionists, I doubt they gain as much traction as OTL (the state of Ontario, as the most populated part of former Canada, may also have such referendums, though smaller and even less successful because they would have assimilated more easily). Their successful example of linguistic autonomy could inspire other parts of the US to try to follow suit though; I imagine there's a greater chance that state governments in the Southwest would recognize Spanish, and a greater cultural revival and prevalence of French in Louisiana (it would help that the state of Quebec would probably build up a lot of historical, cultural, linguistic, and eventually official ties with them). In all, a US that includes Canada would likely grow into a more diverse and multicultural country than OTL, with at least three major languages, a region with a majority Catholic population (at least initially, they'd probably grow out of it like OTL) resistant to attempts to convert them, and a greater respect for the autonomy of its various population groups as a whole.
Quebec in the modern day would probably look fairly similar to the way it does as a province in OTL; a culturally and linguistically unique state where French is the dominant language, and larger cities (where immigrants concentrate) are bilingual to some extent. Independence referendums might happen every now and then like in OTL, though given that the US has had the example of the Civil War as to what happens to secessionists, I doubt they gain as much traction as OTL (the state of Ontario, as the most populated part of former Canada, may also have such referendums, though smaller and even less successful because they would have assimilated more easily). Their successful example of linguistic autonomy could inspire other parts of the US to try to follow suit though; I imagine there's a greater chance that state governments in the Southwest would recognize Spanish, and a greater cultural revival and prevalence of French in Louisiana (it would help that the state of Quebec would probably build up a lot of historical, cultural, linguistic, and eventually official ties with them). In all, a US that includes Canada would likely grow into a more diverse and multicultural country than OTL, with at least three major languages, a region with a majority Catholic population (at least initially, they'd probably grow out of it like OTL) resistant to attempts to convert them, and a greater respect for the autonomy of its various population groups as a whole.