Yes, you can. Those are called "Facebook Requests" and we have a full guide on
Facebook Requests for Joomla
using JFBConnect.
One thing to note is that you will need an SSL certificate on your site. That is a requirement of Facebook due to some of the sensitive information about the user sending and responding to requests. We are looking into some ways that don't require an SSL certificate in the future, but for now, you must have one. If you don't, we have another guide on
installing SSL certificates for Facebook
.
Hope that helps,
Alex