When a marriage is annulled, it’s as if it never happened. All the effects of the marriage are annulled. A marriage may be annulled if the procedural and substantive conditions of marriage have not been respected. Grounds for annulment may include: the marriage was not solemnized by a person authorized by law, a marriage certificate was not issued, one of the parties was too young or did not give free consent (forced marriage), consent was vitiated, etc. In Quebec, a marriage is presumed to have been celebrated in good faith, which means that it is up to the spouse who wishes to invoke the bad faith of the other spouse to prove it.
Note that a religious annulment does not affect the legal validity of the marriage. Consequently, a religious annulment is not equivalent to a legal annulment.
For more information, please see Annulment or dissolution of a marriage | Gouvernement du Québec (quebec.ca) and Annulment of a marriage (gouv.qc.ca)
If you have no legal proof of your marriage, you are not considered legally married in Quebec. You therefore have no reason to request an annulment of your marriage. Note that annulment is distinct from divorce, because annulment establishes that the marriage never existed, whereas divorce recognizes that the marriage existed but is now over.
For more information, please see http://www.cic.gc.ca/english/resources/manuals/op/op02-eng.pdf