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 Tony Abbot 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. (If the governor-general refused, the queen could appoint a new governor-general.)
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))
Also in practice, the monarch usually only appoints a governor-general that has been nominated by the prime minister.
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.
Section 61 of the Constitution says "The executive power of the Commonwealth is vested in the Queen and is exercisable by the Governor-General as the Queen’s representative [...]".
Prime Minister Robert Menzies had to pass a Parliamentary standing order specially for Queen Elizabeth to be able to open Parliament when she visited Australia in 1954 because he received legal advice that, based on the above wording of the Constitution, that power was exercisable only by the Governor-General, not the Queen. So, for the Queen to open Parliament while she visited, a standing order had to be passed through Parliament: "When Her Majesty the Queen is present in Australia and intends to indicate in person the cause of the calling together of Parliament, references in this chapter to the Governor-General shall be read as references to Her Majesty the Queen." Note that this standing order relates only to the opening of Parliament, and is only effective when the Queen is present in Australia.
Constitutionally, the Queen can't exercise the executive power required to dismiss the Prime Minister - only the Governor-General can do this. The Queen can instruct the Governor-General to dismiss a Prime Minister (and, it's worth noting that there's no constitutional or legal requirement for the Governor-General to follow the Queen's instructions), but she can't dismiss the Prime Minister herself.
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).