Firstly, there was provincial birth and baptism registry (and there is none nowadays) for Quebec.Databases of BMD have been produced for the period of 1608 to 1799, but nothing after that.Repertories have been made for less than 10 percent of the parishes.
If you know the DOB and the name of the town or parish, it is possible to access the church records via the digitized records for the early part of New France until 1940 for Quebec.Copies of the actual record can also be produced.
The mother probably delivered in a hospital for unwed mothers. I would think that the birth records of those children do not reveal all the data normally comprised in a birth/baptism record.
There are agencies that help people find the biological parents.More info is available nowadays than in the past.In your case the over 90-year period since the birth should work in your favour.
If you get the DOB and place of birth, I should be in a position to help.