I have certain links/information I want visible only to people who like a specific Facebook page. Is there functionality that will allow me to do that?
I am aware of the fact that users can like and unlike my page. I am fine with that.
I am sorry to say this but what you are asking is clearly not possible. Facebook does not provides.any such feature. Also, pages does not comes under Facebook API so there is no way you can program it for yourself.