No, she can't. Prime Ministers are voted in by the public along with the Party they are leader of. Through this process they essentially have ultimate power and cannot be "fired". They are, however, accountable , through Parliament, to the electorate who voted them in, and can be forced to resign where ideological alienation from the government they were voted in to lead becomes apparent- as happened with Margaret Thatcher in the early '90's (and may well happen to Tony Blair sometime in the near future, but I'll leave that one alone!) or if their inability to govern becomes apparent. Prime Ministers are not "fired", but "asked to resign", and they only do so under public or governmental pressure. In fact, when a PM loses an election it is the usual procedure for both old and new PM to drive to Buckingham Palace and for the old PM to hand his resignation to the Queen before the new PM can take up his role. Just for the record I have no idea what would happen if he refused to do this. The majority of the conventions of British government have built up around historical precedent, with no specific written rules, essentially there is no reason or law to say that this is the way it has to happen, it just sort of does and generally no challenges are made to the status quo. I realise I may have gone off at a tangent here, but considering the oddness of our political constitution compared to the rest of the world, this kind of question is quite difficult to give a definitive answer to, particularly to someone who doesn't live here, I hope this was at least some help.

Essentially there are certain prerogatives that the Sovereign has. These are commonly known as the Reserve powers, and two of those powers are the power to appoint a Prime-Minister, and the power to dismiss a Prime-Minister. While HM has never exercised this in a constitutional crisis, her representative in Australia, the Governor-General did exercise those powers in the 1970's. The powers of the Governor-General are precisely the same as the Queens, and his decisions were held up as being constitutional albeit unpopular, so yes the sovereign can dismiss a Prime-Minister if there is a constitutional crisis, or for the will of the people. Other reserve powers she holds is the right to refuse a dissolution of parliament, the power to force a dissoultion of parliament, and the right to veto legislation when given to her for the royal assent. It would however be something that HM would most likely not exercise. The convention that she doesn’t exercise it does not however mean that the power does not exist. While she is largely a figurehead, she does retain some power, her use of it could create major constitutional headaches. In short however, yes she could fire the Prime-Minister.