An person born in Canada gains Canadian citizenship with the sole exception being a child whose parents are in Canada with a diplomatic visa. Having a child in Canada does nothing to give their parents any right of residency and they must still leave when their study, work, or temporary residency (tourist) visa expire. Children born in Canada receive a provincial (or territorial) birth certificate. This is proof of citizenship.
However, whether the mother is permitted to have a child in Canada depends. Anyone applying for a study, work, or temporary residency visa must submit medical information and typically has to submit to a medical exam. A pregnancy, without a Canadian parent, may cause Citizenship and Immigration Canada to require travel medical insurance (which few pregnant women can get with a pre-existing condition), specific doctors letters, and/or proof that the person will be outside of Canada long before the due date. A pregnancy also makes it unlikely one would qualify for a study or work permit, and would dramatically increase the financial requirements for someone with a pregnant spouse would need to show applying for a study or work permit. Canadian Border Services may also require doctors notes, proof of leaving Canada, medical travel insurance, etc. or deny entry to non-Canadians entered Canada on medical grounds to even visa-exempt visitors.
Finally, even if you end up in Canada pregnant... you had better has some type of medical insurance. Bills for having a child in Canada can be extremely high if you don't have a provincial health plan -- which is not available to temporary residency (tourist) holders and is often limited for study and work permit holders. Hospitals could easily contact CBSA if they suspect you can't pay.