Do you mean a link between the stage of the moon and when parturition (giving birth) actually takes place? Interesting thought but probably not. I study reproductive ecology and I've never come across anything like that in the literature. If a baby is carried to term the timing of its birth is generally some nine months following conception, regardless of the given lunar cycle. If it were the case that the moon did influence the timing of parturition and more babies were born under a waxing gibbous moon, for example, then it would be particularly risky strategy for a fetus that has reached full-term say during the waning crescent to wait around in-utero for the next first quarter phase since there are complications involved for fetuses that are born pre- or post-term. There comes a point when the energetic demands of an overdue fetus start to exceed what the mother can provide in-utero, and so there strong stabilizing selection pressures at play that influence the timing of human births. Also, we know that (contrary to popular belief) there is no link between the lunar cycle and a woman's menstrual cycle, either.
I know childbirth has absolutely nothing to do with the moon. Childbirth is brought on through hormones.