No
Canada is a sovereign country, with its own constitution, government and armed forces.
"Canada Day" is the national holiday celebrated on 1 July, commemorating the date of Confederation in 1867. It has nothing to do with "independence," as that word implies some sort of break from an oppressor. No, Confederation simply distinguished "British North America" as its own Dominion of Canada. Britain was then and has always been our closest relative.
You may be confusing the British Commonwealth with the United Kingdom.
Canada IS a constituent member country of The British Commonwealth. The Commonwealth is a collection of sovereign countries scattered all around the world that at one time in history were colonies of The British Empire and who have chosen to maintain a cordial and sympathetic relationship with Britain, and, more specifically, with the British monarchy. (It's odd, in a way, that the U.S.A. is not a member of the Commonwealth, as you always welcome and celebrate visits to your country of members of The Royal Family. Recall the mania that followed Princess Diana wherever she went in the U.S.)
Another thing that may be confusing you is that Queen Elizabeth is the queen of all commonwealth countries. Therefore it is correct to say that Elizabeth is the Queen of Canada.
Britain does not own the lands of Commonwealth countries, except in a strictly legalistic, fictive sense. In Canada, where there are vast tracts of unoccupied land, we denote those lands as "Crown" lands. It's just an archaic use of the words, The Queen does not in any way own Canadian lands.
To further help you understand this fuzzy relationship, remember this: even in England the Queen's most vital role is not to govern, but to operate the institution of The Monarchy as a symbolic tie to a thousand years of national history, largely "remembered" by much majestic ceremonial activity, and to stand up for All that is Good and Right; to be the country's top diplomat and to discretely advise and be advised by the prime minister. Her Majesty is never involved in partisan politics, no matter what her personal opinions on issues may be.
The Queen's role in Canada is largely the same, though at one remove, with a Governor General of Canada standing in every day to represent Her Majesty.
One final note: a country's membership in the British Commonwealth is voluntary. The U.K. can not compel any former colony to join. On the other hand, the Queen may expel a country from it. South Africa under apartheid was made unwelcome.