Disable Saving/Exporting Results of TSQL Query in SSMS?
No, there is no way to handle this from the database/instance, unless you just deny them access to the database at all.

SSMS is a client-side tool, and the things that it can do with the data you've already let it consume are outside of server-side controls. Even if you could take those menu items away, there is always the PrintScreen button.