It depends what you mean.
If the question refers to the domestic pigeon or its wild ancestor - columba livi - that species is not native, but several others are.
Native pigeons in Australia include:
No. Cheetahs are not native to Australia.
No. Goldfish are not native to Australia.
yes the brolga is native to Australia
No. Otters are not native to Australia. Apart from the occasional zoo, there are no otters in Australia.
The boronia is a plant that is native to Australia.
no, grey hounds arent native to Australia
Dandelions are native to Europe and Asia, but not Australia.
No rabbit of any kind is native to Australia.
Sheep are not native to Australia - they are an introduced species.
Yes. The koala is native to Australia alone.
Kangaroos are native to Australia.Tree kangaroos are native to both Australia and New Guinea.
Wombats are native to the country of Australia.