The Rema mentions a minhag not to go to the mikva Friday night (and Poskim add also motzai Shabbos) if she is not going to the mikva on the earliest possible date. Often women after childbirth postpone going to the mikva until they are fully healed, even if they stopped bleeding earlier. In any event, even if the woman is going to the mikva late, the Rema does not endorse this minhag and says one should go anyways, unless the city or the individual has a specific well known minhag to be machmir. The Poskim write that if one must inquire whether or not this is the minhag, it is not considered "well-known" and one should not be machmir.
So there is some basis to what you heard, but it is not the accepted practice.