We aren't able to turn off the default notification that "Your permissions in Greenhouse have been updated". This is a default notification that informs the user of any changes to their account.
However, it would be dependent on the where the changes are taking place as well. There are two locations in the system where you may update user permissions: the Hiring Team page on a job and the User's profile page.
For the Hiring Team page, any updates to permissions will notify the users by default.
For the User's profile, there is a checkbox that you can uncheck to prevent users from being notified of permission changes. You'll want to uncheck this box and save prior to making changes to job based permissions. See below: