A permanent-resident visa holder with a current driver's licence from another country is permitted to drive for the first three months after arrival. After three months, you will need to hold the appropriate driver's licence for the state where you are living. In Australia, drivers' licences are issued by state and territory governments. Depending on the state or territory, you should contact one of the following:

State / Territory Organisation
New South Wales Roads and Traffic Authority (RTA)
Victoria Vicroads
Queensland Queensland Transport
South Australia Transport SA
Western Australia WA Department for Planning and Infrastructure
Tasmania Tasmania Transport
Australian Capital Territory Rego ACT
Northern Territory Northern Territory Transport Group