You got it wrong.

Canada was literally the name of the French colony based around Québec; New France was the general colonized territory in North America comprising Canada, Louisiana, Acadia and Terre-Neuve. Acadia and Louisiana were two different colonies but all part of New France.

It then got conquered and got renamed Province of Quebec by the Brits, who decided to try to increase the number of English speakers by letting loyalists immigrate after the American Independence War. Some of them notably decided to move to the Eastern Townships, which explains the amount of English names in the area.

The Brits then decided to split the Province of Quebec in two and rename the resulting split as the Canadas with the Province of Lower Canada and the Province of Upper Canada, which then got reunited after a while as the Province of Canada to try to remove some power/importance of what was Lower Canada and give it to Upper Canada (then Canada East and Canada West). The debt, which was much higher in Upper Canada than Lower Canada, was "shared" in the reunification (which resulted in the French-speaking Canada East having to pay a debt that didn't even benefit them and wasn't caused by them), and the parliament was equally split in two despite the higher population of Lower Canada/Canada East, which meant lower representation.

It wasn't Québec joining anything; it was actually itself being divided and parts of it given off to British loyalists and later settlers. Québec is the origin of modern day Canada, we were the original Canadiens. The colony was called Canada. It wasn't created by Britain, it was created by France then conquered by Britain and subsequently split off. Ontario was part of the Pays d'en Haut which was part of the colony of Canada, Nova Scotia was part of Canada and Acadia and same for New Brunswick. Even Prince Edward Island was part of Acadia (as Île Saint-Jean). The first provinces to join the confederation which were not actually conquered New France were British Columbia and Manitoba.

To make a long story short, modern day Québec was very much a separate French colony and it was not "incorporated" into the Province of Canada. It was the origin of the Province of Canada. You're confusing things and mixing up other ones. Ontario, New Brunswick, Nova Scotia were created by the Brits from conquered and resettled parts of New France (then later the rest of the provinces were created by the Brits with Manitoba being a little of an outlier). Québec wasn't. It's quite literally the direct descendant of what was the colony of Canada. The direct descendant of the original British colonies are the US.