I've never heard a specific idea about women in labor praying for others.
However, there's a concept from the gemara Bava kama 92a:
דאמור רבנן כל המבקש רחמים על חבירו והוא צריך לאותו דבר הוא נענה תחילה
A Person who needs something, and prays for a friend who needs the same thing, the one praying is answered first.
So based on this, it would make sense ...