Is it mention anywhere in the Midrash or the Mishna or the Talmud or the Gemara or other Rabbinic literature that when moshiach comes people will no longer be able to have children?
While I'm not sure that's entirely accurate, what you probably heard is something based on this quote in the Talmud, Yevamot 62a:
רב הונא אמר קיים משום דרב אסי דאמר רב אסי אין בן דוד בא עד שיכלו כל נשמות שבגוף שנאמר (ישעיהו נז, טז) כי רוח מלפני יעטוף וגו' ורבי יוחנן אמר לא קיים פריה ורביה לשבת יצרה בעינן והא ליכא The Gemara clarifies the reasons for their opinions: Rav Huna said he has fulfilled the mitzva due to a statement of Rav Asi, as Rav Asi said that the reason for this mitzva is that the Messiah, son of David, will not come until all the souls of the body have been finished, i.e., until all souls that are destined to inhabit physical bodies will do so.
Which seems to mean that one necessary (though not necessarily sufficient) criterion for moshiach to come is that a certain number of humans have been born. That's not exactly your quote, but I see how someone could have creatively read it that way.