That depends on a number of things. No, you can't connect to the Internet from an antenna. But
1) What's on the other end of that TV cable? Do you have cable TV? Is your cable TV provider also your Internet provider? In that case, the cable TV provider may be able to install a modem in that room.
2) Nothing is on the other end. You have a coax cable that runs from where your Internet source is to this room, and you want to push Ethernet over it. There is hardware that can do this.
3) What exactly did you mean if those are both wrong?