After running in the Exchange Shell
 add-MailboxPermission "user X" -user UserY Fullaccess

to give userY the ability to access UserX's mailbox, is there anyway that User X can tell that User Y now has full access?
['https://serverfault.com/questions/244842', 'https://serverfault.com', 'https://serverfault.com/users/73606/']
This is not detectable in Outlook, OWA or any other Exchange client, which is how all standard users usually access Exchange; however, a skilled user (or an administrator) *could* detect it by looking at the actual permissions on the AD user object or on the Exchange mailbox; this requires no actual console access to the Exchange server or to DCs, it can be accomplished with administrative tools running on any client system.
No, unless UserY alters the content in a way that can be noticeable.