No, this is not permitted. The Marriage Act 1961 is very clear in stipulating that a marriage is "... the union of a man and a woman to the exclusion of all others ...".
However, it is possible for a Marriage Celebrant to act as a generalist celebrant and conduct a love commitment ceremony for you and your partner. This is permitted provided the ceremony does not purport to be a legal marriage. In such a case, your Marriage Celebrant would prepare a Certificate of Commitment for you and your partner to be signed by yourselves and your witnesses during the ceremony.
Andrew Sinclair Brisbane Celebrant