Where is the relay and or component that operates the keyless entry remote?

Depending on how many codes are already in the memory, depends on if you can program another fob. The old codes might need to be erased before you can add another. One procedure is to enter the new fob 4 times in a row. There is no relay to change, it runs on it's own module.

Double-check that the dealer gave you the right fob. I know that there are at least 2 identical-looking fobs available, and they are not interchangeable. I bought a used 2001 QX4 and bought a fob on ebay, but it acted just like yours. Then I learned that there were two different fobs, but I don't recall the model year that the fob changed. On ebay they specify this model year or VIN number. After I bought another older fob, it programmed and worked just fine.