If you want to use Facebook Messenger module - it is required to "whitelist" your shop's domain in "messaging" settings on your fanpage. Below you can find detailed steps where I explain how to do it.
open your fanpage on facebook and go to "settings" page,
please open your fanpage on facebook
go to settings tab:
then open "advanced messaging" section
in field "whitelisted domains" type url to your shop (url of shop where you want to run messenger chat window)