Well technically the Monarch and Parliament have disagreed in the past, but there was a civil war over it (Parliament won). So it's tricky how they would settle it now. Though Parliament (in UK) has the right to depose the monarch and get someone else in (Glorious Revolution, where they got rid of the Catholics), so if the queen refused to sign a law, they'd get someone else in.
Funny - in Australia and New Zealand we have Governors General to represent the Queen, and they're also given the final say on whether a law is passed. In New Zealand the people I've talked to have considered it purely ceremonial, but in Australia 1975 that viewpoint was refuted when the governor general dissolved the parliament.
See http://en.wikipedia.org/wiki/John_Kerr_(governor-general) for more.