Canada doesn't have birth certificates. They are a provincial, rather than a federal thing. The provinces also have very different regulations at the moment. So it might be enough, depending on where you were born.
Currently, Ontario is the only province that allows you to change your gender marker on your birth certificate without surgery. All the others require "some form" of surgery. Quebec is arguably the most stringent in regards to what surgeries they accept.
Note that birth certificates are the hardest thing to change. Anything at the federal jurisdiction: passports, social insurance numbers, etc., can be changed with just a therapist's letter. As can the heath cards and driver's licenses in most provinces.
The biggest struggle with getting things changed in many provinces can simply be that very few Service clerks have any understanding of the rules for gender marker changes. It's a lot easier to present them with documentation and tell them to fix it, than to ask them how to get it done. If you ask, they are likely to give you out-of-date or incorrect information. If you tell them to just do it, they are more likely to accept x thing as sufficient. With the exception of Quebec, which is just more difficult all around.
