Yes they can.

I had whooping cough when I was pregnant with Anelise, and my children had it also and they're all fully immunised.
The thing that most people don't realise is that the 'whoop' isn't the sound of the cough... it is a normalish sounding cough but the whoop happens at the end when you're trying to get your breath back (if that makes sense), I always expected that it would be a whoopy sounding cough, but it is the sound at the end... it is an awful thing to get and I like to think that my kids didn't get it as severely as I did as their immunisations are up to date whereas I needed a booster.

Fingers crossed your DD doesn't have it