You have to pick:

A baby porcupine's quills are soft when they first come out and don't have barbs yet so this option isn't as bad as it may initially seem. Female porcupines manage it without injury, after all.

I was imagining an adult porcupine.

Well that's just illogical. :P Still, as long as it's not a breech birth it should be doable. And the quills would have to be soft even for an adult birth or you wouldn't survive the pregnancy to get to the birthing part.

Its maaaagic. One second your fine, then, regardless of whether you are male or female.. BAM! you are giving birth to a fully formed, adult porcupine!

It makes a difference if its clumping litter. Clumping litter you could just choke down the clumps without tasting them. But non-clumping you'd have to taste more cat pee.

Also baby porcupines are smaller than human babies therefore have plenty of room to come out. Assuming you are female.

@errorblankfield: See my comment below. Its not a baby porcupine. Its an adult. :{