There is a messaging link that is next to the friend request link on the page of every Facebook user.
If you want to know how you can remove the messaging link from your Facebook profile to keep people who are not on you’re friends list from messaging you, then keep reading as this post is for you
How to Hide the message Icon from my Facebook profile page from people who aren’t my friend
Unfortunately, this feature has been removed recently. Those who had it set to “no messaging” prior to the change will still continue to not have a “Message” button, but now the only options are “Basic” or “Strict” filtering, which can be found in Settings (the ‘down’ arrow on the upper dark blue band, next to the lock symbol will have a drop-down menu) and then “Privacy” in the left hand side menu > “Who can contact me?”, then “Whose messages do I want filtered into my Inbox?”