I'm not 100% sure either way, but last summer, one of our other dogs (not an APBT) got a rash on her belly, and by the next day (she's small, so I'd picked her up several times by then), I noticed a similar looking rash on my arm where she'd been sitting when I picked her up. The rash was definitely poision ivy, and looking at hers, they were identicle. So, given my own experience, I'd guess that some can get poison ivy, though I don't have any real evidence aside from that.
The stuff on my dog looked like poison ivy, had the little heads and all. If your dogs' doesn't resemble it, maybe it's just irritation from running through the bushes, or a contact allergy.