No, although there are lots of newborns around here, none are canines. This, instead, is a figure of speech.
You know how Bart Simpson would say, "Don't have a cow, man?" Well, no one here -ahem- ever WATCHED The Simpsons, because Mom and Dad didn't allow it, so we had to guess at it's meaning. Once we figured it out, we improvised.
"Having puppies" is the same as throwing a fit, having a hissy, losing it, freaking out, well, you get the idea. Along the same lines comes "having kittens." Having actually HAD four children, I had a hard time even thinking about having a COW...
Once I was with some friends who hadn't heard me speak that way before. When I said to my daughter, who was also there, "Yeah, we could do that, but your dad would have kittens," one friend said, "Have WHAT?" So we had to explain the whole family word thing to her, which was as fun as it gets for Lutheran homeschooling moms. Not really, but there was no chocolate or red wine in the store we were in, so the word game had to suffice.
Now, along the same lines, it will cause you to have puppies if you should say, around our house, "I'm just kidding." That will cause someone to look behind you and say, "I don't see any goats???" You will realize that you are being "had," and it will vex you to the point of, well, you know.