How can I change my Messenger password?

You need to changed it via Facebook

To change your password on Facebook if you're already logged in:
  • Click in the top right corner of any Facebook page and select Settings.
  • Click Security and Login.
  • Click Edit next to Change Password.
  • Click Save Changes.

If you're logged in but have forgotten your password, follow the steps under Change Your Password then click Forgot your password? and follow the steps to reset it