Once the Facebook page(s) and Instagram accounts are integrated within Birdeye with all the permissions, the Facebook/Instagram messages should be visible in Birdeye's Inbox.
In case the Facebook/Instagram page is receiving new messages, but these Facebook messages are not coming into the Birdeye Inbox, you need to follow the below troubleshooting steps to resolve it.
Go to the Facebook page and click on Facebook Page Settings.
Under Page Settings, click on 'New Pages Experience.'
On the next page, click on 'Advanced Messaging.'
Under the 'Handover protocol', configure the 'Messenger receiver' and 'Instagram receiver' as Reviews App.
This will resolve the issue, and the messages will receive under Birdeye Inbox.