I would suggest you reframe the conversation from quitting to taking a break. It doesn’t have to be black and white. If your child is looking at it that way, fine. It’s normal to be more dramatic about things at that age. But as the adult in the room it’s probably worth a conversation with your child about the pros and cons of taking a break. Why does she want to “quit” and why do you want her to not “quit”. Is there middle ground? Does it really matter if she “quits” for a semester? And what’s wrong with her falling behind and having to work hard to overcome her own decision? Can you have confidence that she’ll do that if necessary? These are things to think through for yourself. As children get older you do need to detach from outcomes. It makes discussion and guidance easier if you are less vested in the outcome and more vested in guiding her in problem solving. All parents learn this the hard way - I did!
And there is always the chance when it is truly her call (and not you insisting) the mood will pass.