My Australian friend says yes. I'm not so sure, considering the monarchy collects no taxes from England, much less a country in the commonwealth. What's the verdict? Can she say "you're fired" and Julia Gillard is out?
The prime minister is appointed by the governor-general, who is the representative of the Australian monarch (currently Queen Elizabeth II). The queen could instruct her representative to appoint a new prime minister at any time.
In practice, however, the governor-general usually only appoints a new prime minister as advised by Parliament or a resigning prime minister.
(In a rare exception to this practice, in 1975 the governor-general dismissed a prime minister whose party held a majority in the House. (Wikipedia))
It's important to remember that Elizabeth doesn't have these powers as queen of England, but as queen of Australia. Legally, Australia is a monarchy, with its own royal family. It happens to be that this royal family resides in England, and that the queen of Australia is also queen of several other countries, but only for historical reasons.
The Queen (or King) of England is a constitutional monarch, a ceremonial figure whose powers are strictly circumscribed by the constitutions of England (or other Commonwealth countries).
In THEORY, the "chain of command" goes from the queen, to the governor-general of the Commonwealth, to the Prime Minister of Australia. In actual fact, the queen and governor general will do what the Australian (or English) Parliaments tell them to do.
So the queen does "hire and fire." But not of her own volition, but as a result of the constitutional process.
The short answer is "she does," (what she is told to do) but "she can't" (on her own).