There are many publicly funded sexual clinics across Australia, they are often free and are staffed by sexual health clinicians. Many AIDS Councils operating in capital cities and some regional centres also provide rapid HIV testing and STI screening for gay and bisexual men using a peer-based model. These clinics all welcome trans guys. Testing locations across Australia can be found here.
The recently opened Equinox Gender Diverse Health Centre in Fitzroy, Melbourne welcomes all trans people for primary health care, including transition-related and sexual health.
A sexual health test can also be provided by a doctor but this may not be free of charge.