No.
If you are a non-married couple who live together with your child, then only one of you can claim the child(ren) and the one not claiming the child does not enter anything at all on their tax return about the child. The “sharing” of child-related credits you may have heard about is only possible between divorced or never married parents who live apart and share custody and who have a written agreement to share the credits. The child’s SSN can only be entered on one tax return. Any other return with the child’s SSN on it will be rejected. If you are a family, then work out how to share the refund between yourselves.