Currently Bitbucket doesn't display git-notes and there are multiple feature requests (BSERV-2759 and BSERV-5450) requesting this feature, but even though git-notes are not supported they can be pushed to the repository and they are part of the clone to users local systems.

However, if you enable the Verify Commit Signature hook, notes can no longer be pushed.

While Bitbucket may not support git-notes, we need to be able to push notes when the Verify Commit Signature hook is enabled because they are used for processes outside of Bitbucket