As a guess, you would need to make sure that none of your route goes through a National Park. Another 2c is to ask the "council" up there. I wouldn't want fido to be put down because of some disease control rules or similar.
I have been up there with my dog no probs. Then again my dog kind of blends in, she was rescued maingie looking camp dog from up this way
Seriously tho if you just traverse through a National park on a public road/track and keep the dog in the car, no worries but don't try camping in a national park with your dog.