Generally, World War I. Per "Global Realignments and the Canadian Nation in the Third Millennium":

To this day, the First World War is considered as the crucible of Canadian National identity. Whereas Canada entered the war as a dependent dominion of England, it arose from the battlefields of Ypres and Vimy Ridge as a fully independent nation, the so-called "Nation forged in Fire".

Canada entered WWI under the the declaration of war by England in 1914, not independently. Roughly 7% of Canada's domestic population ended up serving in the Canadian Expeditionary Corps, first under English commanders in separate divisions and culminating in Vimy Ridge, fighting with all four divisions under a consolidated Corps under Canadian leadership.

Canada ended up joining the League of Nations as an independent country in 1919. During the Chanak Crisis (Turkey in 1922), the Canadian prime minister waited for the domestic Canadian parliament to determine the Canadian response, not allowing the British Prime minister to determine Canada's commitment. Eight years later, and the Canadian government was not acting on behalf of Canadian interests, not England's.

There is a whole lot more to it, especially the conscription crisis of 1917, where the English-speaking population supported the wartime efforts and the French-speaking did not. It was another rift in the ever cracking dyke that is the relationship between the two populations in Canada.

WWI is also when "In Flander's Field", one of the best known Canadian poems, was written. It is where the tradition of wearing poppies on Remembrance Day comes from.

The first use of the term 'Canadian' as an ethnonym is dated to the late 18th century, and is commonly credited to Prince Edward, Duke of Kent. From Prince Edward, Duke of Kent: Father of the Canadian Crown by Nathan Tidridge:

Hearing of the violence, Prince Edward hurriedly left Quebec City and headed for the scene of the riot at Charlesbourg, roughly seven and a half miles away. Doctor William James Anderson [...] wrote about the event in his 1870 biography of the Prince:

Prince Edward hearing of it, hastened to Charlebourg, and thus addressed the rioters, in French: "Can there be a man among you who does not take the King to be the father of his people? Is there a man among you who does not look upon the new constitution as the best possible one both for the subject and the Government? Part then in peace; I urge you to unanimity and concord. Let me hear no more of the odious distinctions of French and English. You are all his Britannic Majesty's Canadian subjects." The tumult ceased, and gave place to admiration and applause.

Edward's statement is significant for two reasons: Firstly, the Prince employed the Crown as a great unifier, an umbrella for the separate cultures that were now forced to co-exist in Canada. Secondly, Edward used the term "Canadian" to refer to both the French and English inhabitants of the country. According to Canadian royal historians Arthur Bousfield and Gary Toffoli, this is the first recorded use of this term, in this way.

In a society searching for common ground and identity, whether knowingly or not, the young prince had reinforced the institution of the crown as a great scaffolding within which modern Canada could emerge.

I think there's two phases to the evolution of Canadian national identity, both equally relevant to your question: the transition from 'French' and 'English' to 'Canadian' — which is a process of the late 18th and early 19th century — and the development of a discrete Canadian identity, separate from its global position as a British dominion. The latter part was developed throughout the 19th century (leading up to and beyond Confederation), and arguably matured — as /u/doublelogin wrote — with the First World War.