Not really. Facebook doesn't share information about who like's individual pages of your site as it's a privacy matter. For Facebook Pages, they allow that as it's something that's done within Facebook.com.
There's an option in the JFBConnect -> Social -> Notifications area to receive an email whenever the Like button is clicked. However, the only information we have about the user is their Joomla information. If they aren't logged into Joomla, the email will just say "A guest clicked the Like button". If they are logged in, the email will have their Joomla username.
I hope that helps explain,
Alex