"In a federal election and state or territorial election does the governor-general have to vote?"

Great question, thanks John! As an Australian citizen over the age of 18, the Governor-General is required by law to have their name on the electoral role and to vote. By convention—tradition—, the Governor-General does not vote. The Governor-General or his Official Secretary writes to the Electoral Commissioner to confirm that the Governor-General won't vote and this is accepted.

Check out this link for more information on the Governor-General.

diagram to help explain In a federal election  and state or territorial election does the governor-general have to vote?