Washing machine shutoff valve won't close
You are correct, they are indeed quarter turn valves. To my knowledge, there is no way to repair the valve in question. There is an alternative other than sweating copper for replacing the washer outlet box, but it still requires opening up the wall.

The alternative being a washer outlet box with sharkbite fittings for your shutoff valves. You'll still have PVC, most likely, for the drain to deal with.

I would recommend hiring a plumber to tackle this particular task.