Traditionally, Quebec is considered to be one of the two most important provinces in Canada, with the other being Ontario. This is unsurprising because Quebec started seeing extensive European settlement before Ontario, not least because of its geographical proximity. In modern times, the importance of Ontario and Quebec have faded somewhat because of the increasing importance of the provinces to the west, but for the time being, they remain the most populous parts of Canada as a whole.
Since Quebec was settled by the French rather than the British, its culture is French Canadian. This has caused serious tensions with the rest of Canada from time to time, as shown by the two referendums that happened in the 1980s and 1990s. In more recent times, the interest in separatism seems to have faded somewhat, as shown by the seeming collapse of the Bloc Quebecois. However, it is hard to predict the future based on limited samples, meaning that it is too soon to assume that separatism won't be making a return to Quebec anytime soon.
For that matter, the weakening of the separatists doesn't mean that Quebec has somehow become something other than French Canadian in nature. Now as ever, there are still significant differences between how it is run and how other Canadian provinces are run. For example, Quebec still decides civil matters by using civil law, which comes from Classical Roman law filtered by the French experience throughout the Middle Ages and beyond, as opposed to the common law that came from the courts of the English kings. Likewise, Quebec seems to have a fierce sort of secularism, which is particularly interesting when contrasted with the strong influence of the Roman Catholic Church that existed in earlier times. At times, Quebec's secularism can be perhaps too fierce, as shown by the recent niqab ban that is being contested in courts.
Overall, Quebec is very Canadian, not least because it has been Canadian before there was a Canada. However, at the same time, it is very different from what most people would expect based on their experiences with English-speaking Canada.