You're using Q2A. Q2A uses plugins for Q2A in the same way WordPress uses WordPress plugins or Google Chrome uses Google Chrome extensions.
You've found a chat solution that includes many components (e.g. it uses MySQL, PHP and JS). That has nothing to do with Q2A. It is not a Q2A plugin. If you want to install that and make it work with Q2A, it will most likely not result in just an out of the box plugin but rather it will require a lot of customizations. It is similar to asking "How can I use a jet engine in my bicycle? Please, provide step by step instructions".
So, unless you want to spend hours in getting that chat system to work (or pay someone to do that for you) it is better to look for an already developed plugin for Q2A and use that. There is this one: https://github.com/svivian/q2a-chat-room
. It might not look as good as the one you like but still can get the work done free of charge and without having to spend hours developing it.