Set Up Messenger On Your Facebook Business Page
Go to your Facebook business page and select Settings from the white bar at the top of the page.
General
In the Settings Menu, under General go to Messages and select Edit 
Make sure the box is ticked to “Allow people to contact my Page privately by showing the Message button. Note, if you get a lot of messages, you should turn this off when you’re not available to answer messages.
Messaging
Then go to Messaging in the left hand menu. There are two sections here, General and Messaging.
General Settings
Allows you to set whether the Return key will send messages. Note: if it’s set to YES, hitting return will send the message, not move to a new line. So if you use Return to start a new line, set it to NO and you’ll have a “SEND” button on your messages.
Response Assistant
Set responses. To Edit click CHANGE button next to the message you want to change. There are 3 messages you can edit:
- Send Instant Replies to anyone who messages your Page
Sends a message to everyone who messages you - Stay responsive when you don’t have access to your computer or phone
Will send a message when you’re away - Show a Messenger Greeting
Will pop-up in the message box when people click the Message button on your page
Messenger Platform
Change setting to SHOW to make your page available in the Discover section of Messenger.
Once you’ve got messages turned on, a message button will appear on your page. Note that you won’t be able to see it unless you view page as a visitor.
For more information, check out The Admin’s Guide to Page Messaging by Facebook. It’s got some great tips and gives instructions on how to setup saved replies for those frequently asked questions.