Is there a way to disable just the popup that appears at the bottom right when receiving a message from someone, without disabling all notifications?
Right now, the only two states available are 1.)all notifications (audio, window flashing, and popups) turned on, or 2.)all notifications turned off.
You have likely already found it, but the toggle is found in the arrow drop-down menu. You can disable notifications for an hour, eight hours, or two days.
You may be able to learn more and/or inqure at the Google Product Forum for Hangouts: http://productforums.google.com/forum/#!forum/hangouts
An existing notifications-related thread can be found in the forums here: http://productforums.google.com/forum/#!topic/hangouts/P8RNjXTEvuc