Based on the provided reference, the answer is no.
The reference suggests that Eve, residing in the Garden of Eden, a conceived sanctuary, would not have menstruated because menstruation would have rendered her unclean. The reference states, "because the Garden of Eden was conceived of a sanctuary Eve could not have menstruated since that would have rendered her unclean. From that it seems safe to presume that neither would she have had a discharge at childbirth."
