That's not something that JFBConnect implements natively. If you want to send that information off, you'd need to develop that code on your own. I don't fully understand what the API you linked to does, but that's besides the point.
My guess is that you could create a socialprofile plugin (or even a standard User plugin), to trigger some behavior in the onLogin functions. In there, you can get the session data from the database and push it wherever you want.
Unfortunately, there's not much we could do to help other than point you in the right direction for files to edit. If you have a developer, we'll gladly answer any questions they (or you) may have.
Thanks,
Alex