You can lookup a relay by its IP address:
How can I glean the fp from TBB 3.5?
The fingerprint of which relay? The relay linked on the Tor page is the exit you used to get to the Tor page. All your connections may not use the same exit node, and until you make a circuit, you have no exit node.
If your Tor listens to a control port, you can connect to it and issue this command:
For all your circuits, it lists the fingerprint and nickname of the guard, middle, and exit node.
Most likely you will have more than one circuit and more than one exit node. You don't know from where you exit until you make a connection, and actually exit from somewhere.