No go on that one Spongebue:

"Travel to/from Australia
Pets are not permitted to travel in the cabin or as checked baggage on flights to/from Australia (exceptions may be made for service dogs such as seeing-eye or guide dogs). Pets may only travel to/from Australia as manifested cargo. Passengers should visit Bringing Cats and Dogs (and other pets) to Australia - DAFF and www.unitedcargo.com to learn more"

Care of United Airlines - Traveling with Pets, Airline Guidelines & Breed Restrictions