if my opponent has no cards in their staging area other than their character card, and i activate Q's ability to make them destroy a card in their staging area, will they have to destroy their character at that point?

Q's ability: F Commit: If your hand size is less than 10, both characters' hand sizes are permanently increased by one and both players destroy 1 card in their staging area.

I know in the 1.9 rules it said a character card may be destroyed, im not sure about now, so will this destroy their character? and if so does that mean the game is over?

Q's ability makes no reference to character cards, so it may not cause the destruction of a character card. It's pretty explicit in the rules..

You may only destroy stacked character cards, not the starting character.

And even that's tricky to do, not many cards will be able to do so, they have to specifically mention character cards other than the originals. Q is not one of these cards, he can't destroy character cards at all.