By internet connection on the coax, are you referring to a DirecTV CCK connected to the line or do you have Cable internet and are referring to that? Short answer to this is no.
There are adapters that can be used to make a coax carry TCP/IP network (read internet) data (here's a link with several:
http://www.netsys-direct.com/category_s/1818.htm). You'd have to disconnect that coax from any other services and find a way to get the ethernet back to your router.
Another option would be to use an RF adapter and hook up composite cables to another receiver. I do this with my receiver located in the Kitchen and it feeds an old SD tube tv in a spare bedroom.
Another option would be if you have Wi-Fi, you could simply ignore the coax altogether. Put the slingbox near a receiver and get an internet connection to it, then use a laptop or Wi-Fi capable desktop/media center PC to access the streaming content.