Not doable with a POD after 1900. You didn't have the concept of Quebec Separatism as we would know it post-WWII that led to much of the modern 'separation' attempts. Republicanism in Canada was also virtually non-existent during this time period.
However, you could see something like a large series of general strikes, work refusals, conscription resistance, provincial government refusing to bow to pressure from Ottawa, and potential overreaction from Ottawa leading to an 'occupation' of Quebec in response to a worse conscription crisis. Though this does require a more organized resistance movement that is exclusive of the general reluctance throughout Canada to conscription. Still, none of this would inspire a 'rising' in WWI.
Though to get something like what the OP would like, you have to have Quebec feel like it has even less stake in the war than it did OTL, a coherent Quebec Nationalist movement existing beyond 1867, and a strengthening of republican sentiment in Canada. The last one is really the hardest to do since it never had widespread support purely because of how Canadian identity developed OTL in direct opposition to its southern neighbor.
Good analysis.
Remember, the division back then was much less between 'Anglo'and 'Franco' as between Protestant and Catholic. So the Irish were definitely on the 'Quebec' side, for instance.
I could see this happening if Alt-WW1 involves Britain at war with France.
Your challenge, should you accept to take it, is to have a separatist republican rebellion in Quebec, comparable in scale, goals and support to the Irish rebellion. Wether it is successful or related to the events in Ireland is up to you.
Eh, the French Canadians don't see themselves as having much connection with France.
Not true. Back then (and to a certain extent, today too) there was a love-hate relationship with France among French-Canadians. Most French-Canadians (especially in Quebec where the vast majority of French-Canadians live in) back then as today, don't feel a connection with the rest of Canada either. They think of themselves as Québécois first, Canadian second if at all.
It's one thing to oppose conscription forcing French-Canadians to fight alongside France but a BIG ANOTHER THING if Canada and France are on opposing sides and French-Canadians are forced to fight AGAINST France.
I see...From what I've read, that's why Levesque was so unusual in that he didn't care about France, but admired the US (unlike other prominent Quebec separatists).