Its my understanding that a botched siege of Quebec City was a major early setback for the revolutionaries at the start of the war. Considering that Quebec was a recent conquest, under the control of Britain with its different language and religion, and the fact that almost the entirety of the rest of Britain's American colonies were flaring up in revolt, and the entrance of Quebec's former mother country, France, into the war against Britain, why did they seem to be content enough with British rule not to cause any particular trouble? Were they worried about being sucked into some kind of radical Protestant new state with a worse deal than they had with the British?
On a related note, were the French hoping to somehow regain their former possessions in Canada when they supported the revolution?
You really hit the nail on the head. The French in Quebec gained extremely favorable terms after the Seven Years War. They were able to remain a Catholic province, retain the code of laws and pretty much have their day to day activities stay unchanged. From the British point of view, it made sense. Any drastic changes would have required a significant occupation force, and the war had really drained the coffers.
The hard core Protestant Americans were not pleased. Their French neighbors were an existential threat since the previous century. Catholics were the devil incarnate, and they were right next door. New Englanders had fought at Louisbourg, Ticonderoga, all for nothing in their eyes.
For the British, the war was a world war, and these concerns barely showed up on the radar.
The folks in Quebec were not unaware of the feelings that New Englanders had towards them. In 1775 there were about 90,000 people in all of British Canada. There were 2.5 million Americans. If anyone should have felt an existential threat, it would have been French Canadians.
Frankly, in the end, the Quebecois did not, as a majority, have the same issues with England that the American colonists had, or rather, thought they had. Bottom line is that the citizens of Quebec City had much more to risk by siding with the Americans than simply remaining with distant British Government.
You can also find more context here u/enygma9753 about Quebec's complicated and perilous situation in the leadup to the Revolution.
New England and the habitants / Canadiens (the inhabitants of Quebec) had long been continental rivals throughout New France's 150-year existence up to the 1759 Conquest. During this time, both sides engaged in alliances with various indigenous groups, motivated by political, economic or defensive reasons -- the Iroquois largely sided with Britain, the Huron with France.
Both sides were regularly involved in what we would now call guerrilla warfare on the frontier, either as proxy wars for the larger dynastic conflicts in Europe in the 1600s and 1700s or as part of the frequent tit-for-tat local disputes over lands, resources and trade. Rival settlements were raided or burned, civilians attacked or killed and atrocities such as scalpings occurred. It should be noted that white settlers on both sides actively took part in such acts too, not just the natives.
I included this context to emphasize that there was no love lost between Quebec and the Thirteen Colonies by 1774-5. Not only due to the virulent anti-Catholic sentiments and anti-French bigotry in New England, but the utter lack of trust the Quebec habitants had for the American colonists. Quebec's settlers were not fond of their new British masters -- but they loathed their New England neighbours.
The Quebec Act did enough to blunt any temptation to rebel actively, but not enough to encourage the habitants to take up arms for Britain.
In contrast, while the Americans extended a formal invitation to Quebec to join the revolution with promises of liberty, there was a lot of mixed messaging. During the brief occupation of Montreal, the American occupiers persecuted Loyalists and harassed Catholic clergy and hardly appeared to be as tolerant and accommodating as they professed. The Patriots' vague offers of liberty lacked the legal guarantees to rights that Quebec already enjoyed under British rule. Benedict Arnold's failed attack on Quebec City also soured their impression of the Patriot cause.
The best choice in their eyes was to let les Anglais and les Bostonnais fight among themselves, rather than taking up one side or other and risking the rights and freedoms they had just won peaceably by law.