The term used for when one character is more interested in children than romance. And also mostly a joke. Generally the name changes from "Deacon Babies" to the character who wants the babies' name, or the character who the first character wants to have babies with, depending on how it's being referred to. The character generally wants the other character's babies. But the children themselves can be referred to by either parent.
EX: If John wanted to have kids with Jen he would want "Jen Babies". However, their children could be either "John Babies" or "Jen babies"
It's escalated quite far from there. I think it's basically a meme.