Hi L.,
I'll bet you she literally has no idea she's even doing this. Honestly, I don't know a Mom out there who isn't ready to bring up their own child. (like me).
She's showing poor *listening* skills: she's just waiting for her turn to talk: you brought up your daughter, so it's a opportunity for her to bring up (brag about) hers (and forget the two of you were even talking about your daughter in the first place).
It's an annoying habit, but difficult to change in someone else. Maybe, first ask how her daughter is doing. Let her talk as long as she needs to, then start up on your sweetie.
If she can't keep her mouth shut for a while about her kid after that... ?
Good luck.
t