Thanks to everyone who replied with suggestions. Unfortunately none of the suggestions worked but I finally managed to figure out a way to resolve it. Find the Google Messages app in the Play Store and tap on uninstall. It won't actually uninstall the app because there isn't a way to get rid of that one. Instead it will uninstall all the updates that app has ever had and that resolved the issue. No more unread message icon! Then tap to install all the updates again and everything is back as it should be.