This is not possible someone would have to "Communicate" to your web server to retrieve the quiz. Being that is said what you can do is get one of them free shells that allow IRC communications, and have a bot come to your channel who is permanent that allows you to trigger the quiz script.
I would personally stay well away from "free shells" especially as a small and decent shell can be used for $3US a month.
Or run a windrop if you use windows Or run an eggdrop or perl bot if you use "nix".
