Ciao Michele,
We would need the Free/Busy error for the affected direction. Assuming it is cloud to on-premises, check the error with Fiddler or OWA F12 (look at GetUserAvailability* as explained above) and check Get-IntraOrganizationConnector and Get-OrganizationRelationship settings in Cloud. And also on exrca.com the free/busy test under O365 for Cloud user to OnPrem user (assuming direction is cloud -> onprem). This will say if pass through is there indeed at least for Basic Auth (what exrca tests now) and other things like ports , certificates. Could be also a TLS issue, I cannot guess, so we need the F/B error message and you can also check the most common errors here: FB_Errors.FixesV6