I suggest contacting your TA or the airline and see what they say. These are really the only people that will know for sure if you will be allowed to board with that birth certificate and if your b.c. is a no go then they will also be able to give you suggestions on what you should/have to do so that you can get on the plane.
Quebec birth certificate expired
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In BC you can get your passport very quickly for an extra charge if you take your i.d. directly to the office.
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My wife was born in Quebec and had to get a new birth certificate . I think it took about 2 -3 weeks for the certificate to arrive.