The Queen doesn't need legal rights, she is the law!
As per Dale M's answer, Her Majesty doesn't meet the requirements to be a citizen of Australia under the Citizenship Act, nor has she ever been granted that status.
Someone else will need to fill in the legal reasons as to whether or not a law like the Migration Act could be enforced against Her Majesty personally. I suspect there is an argument in there about the Crown generally not being bound by statutes. In practice, I can't imagine anybody at the Department of Immigration trying to turn Her Majesty back at the border (although this would be an interesting step towards a republic...).
In any case the Royal Family and members of its entourage have the special status of not needing a visa when they visit Australia: Migration Regulations 1994, reg 2.40.
This guy did most of my research for me: http://andrewbartlett.com/does-the-queen-need-a-visa-to-enter-australia/