do you have a TV monitor in your room with analog composite input ? Or analog component input ? Bring up color bars in Final Cut Pro, and make sure that you can get these color bars from the AJA I/O Video output into the TV monitor composite input. If this works, that means that you are probably getting an SDI output from the I/O as well.
You have not told us what the TV monitor looks at. Some people have the I/O go into the VTR, and the VTR goes to the TV. I have no idea if you have a patch bay system or router system in your room, or if this is a simple small room, with the I/O and VTR and TV all in one room.
If you cannot get color bars onto your TV, simply open up the Pro I/O control panel, play back a tape thru the I/O, and see if you can see it on the TV monitor (forget FCP for now - just try this test). If you can't see an image thru the I/O with the control panel, then something else is up.
There are many reasons why you could not send an SDI signal into your VTR (often cabling), but you must tell me more about your room, for me to give you more suggestions. If your machine is simply patched out (or you dont' have SDI INPUT selected on your VTR), no suggestion is going to help you.