No. Private health insurance does not cover visits to see a non-GP specialist in their consulting rooms.
Instead, visits to a non-GP specialist in their consulting rooms are covered by Medicare.
If you are enrolled with Medicare, it will pay some or all of the cost:
However, if you have Hospital Cover and the appropriate Hospital Cover tier (Basic, Bronze, Silver or Gold), private health insurance may cover some or all of your costs of treatment or surgery by the non-GP specialist in a public hospital or a private hospital.