The Verdict: Unlikely
As far as we know, the ADF does not yet recruit permanent residents of your nationality. Check the current eligibility list on ADF Careers, and look at ordinary citizenship by conferral meanwhile.
Recruitment opened to New Zealanders living in Australia in July 2024, and to permanent residents from the United Kingdom, the United States and Canada in January 2025. The government has said it may widen this, so the list can change.
Ordinary citizenship needs four years of lawful residence in Australia, the last twelve months as a permanent resident. That route is open to you whatever your nationality.