Yup, it's possible; you can do it multiple ways;
One route is to use official websocket plugin, and write-up a server in python, node, or whatever suits your need.
Other is to use socket.io plugin, there are several versions available on the forum, latest stuff was added by aceofspades, i think. The route is the same, you'll need a server in some language that has bindings for socket.io.
Other libraries you can use are now.js, sock.js, even pubnub.com.
For this application, i would probably use official websocket plugin, it's most stable one, and i think you can do everything with pushing text messages around connected clients!
Edit: Actually, for a quiz there's no need for websockets, you can do it with Ajax!