I would discourage you from even looking for a chat feature as I don't think you will find one that integrates with QuickBase data. You will save yourself a lot of trouble to just use an existing chat program in a separate window. There are a lot of chat clients that can be embedded into a website using a fragment of HTML code but they are unlikely to be able to integrate with QuickBase data automatically via an API unless you are going to include your own an external server.
Completely agree with dandiebolt_QB_Pirate. I have a client who embedded JomSocial into the app homepage so franchisees could chat within the application.
I am not sure I understand your comment or why you completely agree with me. Isn't JomSocial a plugin for the Joomla CMS? I think the OP was asking for embedded widget to a QuickBase dashboard or other location. Further I think by embedded the OP wanted to not just chat but to access QuickBase data without cut and paste. Please elaborate on how you used JomSocial.