The Queen has all kind of royal titles, such as Queen of Australia and Canada and various other places, but "Queen of Great Britain" is not one of them.

This would be because officially there isn't any country with that name - it's the name of the island on which England Scotland and Wales are perched. "The United Kingdom" is the term used, to include those countries, and also (for the time being) Northern Ireland. (But not the Isle of Man.)

All very bizarre. Including the fact that she's officially called Queen Elizabeth II, but there's never been a Queen Elizabeth the First in most of the places she's supposed to be queen of, such Australia or Canada or Scotland...

The Scots used to go round blowing up pillar boxes (post boxes)saying EIIR, for that very reason, but they seem to have given up doing that.