Facebook, Google, and other services which use short codes to send SMS messages will not be able to send messages to Twilio phone numbers. This is because short code carriers have arrangements to exchange messages with mobile phone numbers only, and Twilio phone numbers are not considered mobile phone numbers.
Twilio's API can however be used to send Facebook messages in Messenger with minimal changes to your Twilio application. This feature is currently available in early access. You can sign up for access here.