I have no idea what would happen. I suggest you try it on a test server, then tell us what happens.
If the quiz does not have a time limit, then it will probably be OK. Everything is saved as the student progresses through the quiz. They will be able to continue when the system comes out of maintenance mode. The problem will be if time expires on a quiz while the system is in maintenance mode.
From my memory of playing with maintenance mode I would suggest nothing bad would happen if you were taking a quiz. But I recommend trying out on a test server to be certain.
Ok, I gave it a try.
When the site switches to maintenance while you're in a quiz attempt and you press submit, you're taken to the maintenance notification (as expected). If you go back to the previous page via browser arrow you can, or rather have to, retake the quiz. The timer and previously entered answers are reset.
If the site goes out of maintenance and you refresh while you're on the maintenance notification page, you're taken to the quiz review page (where you would have ended up without maintenance mode), but the timer keeps running in the meantime.
So if you only have one page in your quiz (page breaks result in the maintenance notification) and your quiz is set to auto-submit when the timer has run out, it's not a big problem.. but it's probably a good idea to add an explanation/instructions to the M-mode-notification, like..
"if you were in a quiz attempt and came to this page when you tried to submit: your progress has been saved and will be auto-submitted when the timer runs out, you may now close this tab or refresh it later..." or something like that, just to make sure people don't panic.
regards,
Seb