Though not an option in the current platform there are few possible avenues.
1. If you would like to see this as an option in a future release I would suggest submitting an idea in the Ideas for Jive space where our Product Managers review top rated ideas when considering new features. (similar idea here: Switch on email notification on Connection stream on Jive Cloud by default )
2. If you are wanting this to be in the users inbox so that they receive email notifications, you should be able to modify the connections stream by setting the email = on to achieve this functionality:
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There is one way for developers to achieve this using jive webhooks.
Using webhooks you can get notified of the event where a user becomes a member of a group.
On this you can associate the inbox (or any other stream) of this user with that group.