Probably just me being nitpicky, but the name of the polity you're discussing was and should be Canada. It was 95% French, and they proudly called themselves Canadiens. Québec, for them, was just the name of their capital.
The name Québec referring to a state is much more recent. Only after Canada had become majority-English and budding Francophone nationalism was looking for a more pleasant identity than 'Low Canadians' and 'French Canadians' did Québec become a new rallying name.
Anyhow, this detail aside. There are a few difficulties with that POD, namely that the Canadians were still quite distrustful of their historical enemies in the 13 colonies. Significant tensions with the British though could change that. If Britain tried to force concessions on religion, that's the easiest selling point. Catholicism meant more than us than even French back then; and if the British had tried to oppress the Church, anything would have been possible. Canada also had significant interest in preserving it's territory in the Ohio valley.
So a perfect storm might be (even worse) British misrule, and enough openness on the part of your founding fathers to offer stronger state-powers in matters of culture and religion (or alternatively an amendment that addresses Canada's concerns in exchange for ratification), as well as guarantees for our economic interests in the valley. In time, simple demographics leave little chance that the Ohio could be prominently French Catholic either way; likely it would become states of it's own someday like in OTL.
North of the current US-Canadian border, however, the prospects of the French would be better in the short term. Without the exodus of English loyalists to Canada, nothing would threaten the Frenchness of Canada from Labrador to the Prairies; perhaps in time Acadie would be recovered as well (1812?). An American Canada this early would likely help cement Franco-American relations and without British occupation, further migrations and contact between France and Canada would occur throughout the 19th century. Ultimately, though, if the US tried to centralize as much as it did IOTL after the civil war, this would have been a problem. While being strongly anti-slavery, Canada would have probably shared with the south the vision of more autonomous states, as is likely in any scenario with strong cultural differences. Assuming it all sticks in one piece, who knows how much this transforms the balance of power and the role of DC in the long run? French Canadians were also much closer to the native tribes of the greater Ohio and Louisiana, with whom they had traded for 150 years, so it could have significant impact on how much the US ends up mistreating the natives as it marches west. If Canada retains control for a time of the Ohio valley, much of their traditional allied tribes would be on their turf, so to speak, and it could well mean greater surviving native populations in the long run.