The Australian government pays for the research study fees.
Who will, in such case, have the intellectual property rights for the research: the PhD student or the university?
Laws vary and I am neither a lawyer nor Australian. But in general, unless the grant or university rules say otherwise, things you create are (or at least should be) your own. Some universities try to make a claim (various places in the world) and you may have signed away some of your rights previously, but in the absence of that the IP is yours.
Check locally for the correct answer, of course. If the rules seem arbitrary or unfair, you can explore, locally, what it would take to counter them.