You might be interested in reading my recent post about French Canadian identity during the Great War. Much of the tension in Canada occurred over conscription, which deeply divided the nation, and has often been characterized as pitting pro-war Anglo-Canadians against anti-war French Canadians, although it was more complicated than that.

In addition to the linked post, I think there are two important points that should also be addressed. First, there was a much larger ongoing debate about the nature of "Britishness" in Canada (and other Commonwealth states). Before either war, and especially during the interwar period, there was a significant effort to make Canada as British as possible. Sarah Carter's The Importance of Being Monogamous and Imperial Plots, just to name two books on the subject, demonstrate that there was a concerted effort to bring British immigrants to Canada and model Canadian society along British lines, and non-British immigrants were classified by their possibilities to become British (before WWI for example, Germans were relatively well liked because they learned English well and assimilated into predominately British-style communities. Groups like Russian Mennonites, on the other hand, faced far more resistance, because they made almost no attempt to assimilate into British-Canadian society, living instead in isolated groups, practicing non-British forms of religion, and speaking still almost exclusively Russian or Slavic languages).

French Canadians were inherently not British, either in origin, language, or religion, which caused a lot of consternation amongst English speakers. Consider the case of Manitoba, which was inducted into Confederation with legal safeguards for the large but still minority French Catholic population. Over the course of about three decades, the English majority was eventually able to completely remove funding from Catholic (i.e. French) schools, make English the only language of provincial government, and even deny the ability of French individuals to have court cases conducted in French (sometimes, the defendant was the only individual to speak French in the whole courtroom, and translators were not provided). This pattern was repeated broadly in most English majority areas. French Canadians were discouraged from moving to other parts of the country, especially the prairies, out of fears that they would develop non-British settlements that would have to be given consideration, however token it may be.

When Britain declared war in WWI, Canada was automatically at war; not until the Statute of Westminster in 1931 did colonial countries have the ability to determine their own entry into war. A lot of French Canadians saw this as another example of the need to be British at the exclusion of all else. After all, Canadian involvement in the war was the result of British colonial ties, not Canadian concerns, and the majority of officers were British until quite late in the war. There was little effort to have French speaking regiments, and French Canadian concerns about the war, many of them legitimate, were dismissed by much of English Canada as further evidence of their fundamental anti-British nature. These, among other concerns including a soldier voting law of dubious nature, are part of what led to the riots in Quebec that occurred when Borden announced conscription.

The second important point is that King was less concerned about French Canadians than he was about his political security; as I'm sure any Canadian will tell you, it's very difficult to win a federal election without at least some support in Quebec, especially in this period when the prairies still had minimal population and therefore seats. He was also, paradoxically, a staunch Imperial British supporter while simultaneously attempting to secure a more independent Canada that did not have to get approval for almost everything from Britain. All this results in an approach of wanting to support Britain in WWII (initially with the British Commonwealth Air Training Program, to minimize financial and personnel commitments) while wanting to avoid accusations of supporting Britain because they were, well, Britain, something that would have lost him a huge number of votes in Quebec. King famously avoided conflict between the predominant English Canadian (British) majority and the French Canadian (non- or even anti-British) minority by simply avoiding dealing with questions of conscription until relatively late in the war.

Two books I would recommend you read for more nuanced understandings are Roy Maclaren, Mackenzie King in the Age of Dictators: Canada's Imperial and Foreign Policies and John English, The Decline of Politics: The Conservatives and the Party System 1901-1920 (this one especially details the issues surrounding Borden and conscription).