According to Facebook, You can't log out of an account on Messenger, but here are some things you can do to reduce notifications, make yourself unavailable or stop getting messages from certain people:
1. Turn off or adjust notification setting
2. Block someone
3. Appear as inactive
4. Switch accounts
5. Control your privacy
Note: If you think you've left yourself logged in on someone else’s phone, you can always end an active session from your Facebook app's Security Settings.

But this trick will help you how to log out of Facebook Messenger