The queen (or king) can abdicate but would not have the power to abolish the British monarchy. That's a constitutional change and would presumably take a Parliamentary act.
If she wanted to abolish the monarchy, she could have spurred that change pretty easily I think.
A hell of a way to end the Monarchy would have been to use the royal prerogative to install an anti-brexit Prime Minister a couple years ago (against convention, but that's the point). Presumably that'd be enough to get them to abolish the Monarchy. And it would have been an fitting end to the Monarchy, a legacy of the previous era, to have the Queen, who held it for the current era, expend it's last bit of power to stay in the EU, which might be one of the top players in the next era.
Well that's how it'd be written if it was a movie at least.
