First of all - Almost everyone wants to quit at some point. That tends to be the nature of children when something is hard. I'm a believer that quitting is not an option when you've started something you've committed to.
However, in the case with my son - each day was a fighting, screaming, crying battle. I don't think he had more than 5 happy practices in the four months he was taking lessons. It was absolutely draining to me - knowing that I TEACH children, why can't I handle my own? When it came down to it, my son never wanted to play the piano, I just put him in lessons because I thought I should. He had absolutely no desire so it was torturous for him. I hope that parents don't force their children to do something they never wanted to do in the first place. With my girls, it was a different scenario - they both wanted to play - and I won't let them quit.
Here are some things to try when someone wants to quit:
1-Make sure they stick with it until the next performance. Sometimes they just need a reminder how good it feels when they perform and everyone tells them how great they are.
2-Ease up on practicing. Whether you are a parent or teacher you can tell when a child is struggling and so assign less (if you're a teacher), or demand less. I almost wrote "expect less," but I don't think you need to expect less in quality or time. Just require less of the child. Often children just have a slump of enthusiasm, but something may spark it again.
3-Don't overschedule children! With lessons comes practicing. So, if someone is in any sort of lessons they need to have the time to practice. If a child is doing sports, dance, church activities, after school activities (not to mention homework - and just plain downtime) is there any time to practice? If there is no time to practice that may be why someone wants to quit. If it's not the lessons you want your child to quit, then you may need to chose another activity.
In short, I don't think quitting is really the best option, unless it's the best option. My son has NEVER asked about piano lessons since the day I called his teacher and said we weren't going anymore. However, 9 months later I hear coming from the living room a choppy rendition of "Dashing Through the Snow" played by my son. Something stuck in that first four months and maybe when he's a little older we'll take another stab at music lessons, but not until we make sure it's something he's interested in committing to.
....and a little side note - I HAVE heard adults say "I wish my mother didn't put me in piano lessons when I was a child."