The answer I had from the Canadian authorities was the following. They are not the final word on what Qatari officials would do but it was frightening nonetheless:

A woman giving birth in Qatar without a father would encounter serious difficulties. The child would not be released until the father had been "recognised" by the local authorities, and possibly could be taken away from the mother, depending on the circumstances. We've had Canadian parents who had children born here having problems to have the child released form the hospital and obtaining a birth certificate until they could prove to the authorities here that they were legally married.

A child born to a Canadian parent is a Canadian citizen regardless of where he/she was born. If your child was born here, he would not acquire Qatari citizenship as only the father can transmit citizenship to a child. Therefore, should you have a child here and assuming that you would be able to overcome the difficulties of having a child "out of wedlock", your child would be in the eyes of the local authorities a Canadian citizen only. You would need to obtain a Canadian passport for the child, and then sponsor him/her as your dependant for a Residency Permit.

If you were to have a child in Canada and come to work in the region, the child would have a Canadian passport and you could sponsor him/her without problems. However, should the child be of schooling age, you could once again encounter problems as the child's birth certificate that would be required would show the child fatherless. Even divorced mothers must provide proof of divorce for matters concerning the children.