Legal Question in Family Law in New Mexico

The short answer is "no."

I don't know what a "traditional marriage" is but New Mexico doesn't recognize a common law marriage, if that's what you're talking about.

Smart thing for this couple to do would be to do a paternity action to get a child custody arrangement including child support and a parenting plan put in place by the court. If she considers the girlfriend a threat of some kind to the child that she can document then she can use the parenting plan and court authority to limit visitation. A new "parent" figure would probably not be considered by the court to be a threat to the child, lots of kids have four or more parent figures. A threat in this context would be some kind of child abuse or neglect, a domestic situation that would include drugs or alcohol to excess, etc. Most courts default to 50% time-sharing absent some reason such as above to limit access.