Your girlfriend will not get PR or gain the eligibility to apply for PR simply by having a baby in Canada.
If she wants to get PR, she could look into applying by herself if she qualifies or you can sponsor her. In order to sponsor her, you either have to have lived together as a couple for 1 year (common law), or get married.
The processing times vary but her being from the US, if you apply outland which means she may visit you in Canada or stay in the US, she may have her PR in around 10 months. If you apply inland which means she must stay in Canada for the duration, she may get first stage approval in about 9-10 months and PR in another 8 months.
The baby will be Canadian if born in Canada but note that if she comes to Quebec as a visitor, she will not have health care and you will have to pay for the delivery yourself. This can easily cost you several thousand $. If the baby is born in the US, the baby will get US citizenship but as you are Canadian, you pass your citizenship onto the baby as well. You would just have to apply for a citizenship certificate for the baby to register that.
My advice is get married and apply to sponsor her. For US citizens, it is usually better/faster to apply outland. She can still visit you in Canada and if she wants to stay longer than 6 months, she can apply to extend her visitor status. She can not officially move to Canada until she gets her PR so do not show up with all her belongings in a U-haul at the border because that is asking for trouble.