Facebook lets you connect with your friends and family via both public comments and private messages. You have the option to disable other users from sending you direct messages by removing the message button from your page altogether.
This tutorial will walk you through the process of doing so.
How To Stop Private Messages on Your Facebook Page
Connect to your Facebook page. Go to Settings > General:
Edit the Messages section. Clear the Allow people to contact my Page privately by showing the Message button checkbox:
Click Save Changes.
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