desperately want to play quake with my son, is there any way to use a controller?

Galaxy is just a client, the ability to use a controller depends on the game itself. My son and I play many games using controllers.

You will need an Xbox controller mapper software, first truly off search:
https://www.rewasd.com

There are plenty of others. Though why you want to is beyond me, it's like playing the game wearing boxing gloves, keyboard and mouse is the best way to play fps games.

Well, Xinput didn't exist when Quake was made, so you'll probably need a converter for that.

Like others have mentioned you'll need a remapper, like:

Pinnacle Game Profiles
Xpadder
JoyToKey
AntiMicro

It's kind of a pain in the neck to play keyboard and mouse games with a controller, but it can be done if you really feel the need to.
Still if you're going to play something like Quake with him, you might as well teach him how to use a keyboard and mouse.