There are several settings you can optimize on your Facebook page to better promote your Messenger channel
Make sure you've set your page to allow messages. Go to your page Settings > General > Messages and make sure 'Allow people to contact my Page privately by showing the Message button.' is checked.
Prompt visitors of your page to send you messages
Facebook will promote your Messenger in a variety of ways by turning on this setting. Make sure you enable this feature by navigating to page Settings > Messaging > General Settings. It's a great way to grow your Messenger audience!
One way is that visitors of your Facebook page will be prompted to send a message, with a prompt like the one shown below.
Make sure you get rid of any conflicts on your Messenger
It's wise to check some of the Facebook's features in page Settings > Messaging > General Settings that allow automation. They may conflict with your Messenger experience.
Disable other platforms if you already have a Messenger experience.
Some apps other than I AM POP may influence your Messenger and cause conflicts. Navigate to your page Settings > Messenger Platform > Connected apps and hit 'Remove'.