Depending on which state you live in, you can register your relationship here:
New South Wales: https://www.bdm.nsw.gov.au/Pages/marriages-relationships/relationship-register.aspx
At least one partner must live in New South Wales. It will take 28 days to receive your certificate after you have registered your relationship.
Victoria: https://www.bdm.vic.gov.au/marriages-and-relationships/register-a-domestic-relationship
At least one partner must live in Victoria. It will take 28 days to receive your certificate after you have registered your relationship.
Queensland: https://www.qld.gov.au/law/births-deaths-marriages-and-divorces/marriage-weddings-and-civil-partnerships/civil-partnerships/registering-a-civil-partnership
At least one partner must live in Queensland. It will take at least 10 days to receive your certificate after you have registered your relationship.
South Australia: https://www.sa.gov.au/topics/family-and-community/births-deaths-and-marriages/register-a-relationship
At least one partner must live in South Australia. It will take 28 days to receive your certificate after you have registered your relationship.
Australian Capital Territory: https://www.accesscanberra.act.gov.au/app/answers/detail/a_id/1694#!tabs-2
Completed on paper application.
Tasmania: https://www.justice.tas.gov.au/bdm/significant
Both partners must live in Tasmania. It will take 28 days to receive your certificate after you have registered your relationship.
Western Australia: N/A for Migration Purposes