Yes Australia has strict rules for dogs as where dogs are strictly not allowed then there will be a sign board. But theres nothing with Relocating your pet. They have great service for that.
I think you are referring to Dog in Public laws in Australia. The post was about importing a dog to Australia.
Do not get confused. When importing any live animal (dog, cats horses, fish) into Australia, there are very strict, complex, sometimes expensive and lengthy procedures the owner must follow.
There are specialist freight companies, accredited by AQIS that can handle the whole process door to door.
A friend of mine recently returned from Thailand with their dog. The dog was sent to Singapore 8 weeks ahead of the owners, stayed in quarantine in Singapore for 8 weeks, then shipped to Australia. When the dog arrived it then spent another 2 weeks in quarantine in Sydney. The poor dog had 6 veterinary exams, 4 blood tests. The total cost (including airfares for the dog) was AUD$8500. Its not cheap. But price and difficulty varies depending on where the dog is coming from.