I know how to set up a QT tank for a fish, coral, or clam but not sure about eels. If they are done like fish you need a tank, heater, sponge filter/or HOB filter, some media to help with the cycle and a few pieces of PVC for them to hide in. You don't use live rock or sand for fish in QT.
If you're in a real pinch you can go ghetto-fabulous with QT and use a rubbermade bin but you'll want a good cover on it for that eel I'd think. And of course don't seal it off completely from air.
I'd personally get the eel out as I said and really investigate how it appears right now. If there is a chance you can treat it, having it in a QT tank will be far better than in the display tank.
Either way, whatever you decide please read more about QT tanks so that your current fish have a place to go if needed as well and any new livestock are placed there before going in yoru display. I learned that one the hard way, see my signature for my new mantra.