Hi Chipperdancing and Welcome to the Forum:
Yes, you can change the thremocouple. There are different kinds but I will refer to the most common one.
1. Look at the gas valve; there should be 2 small lines that look like aluminum, one runs to the pilot itself and the smaller one is attached to the thermocouple.
2. Loosen the nut on the smaller line, look inside the burner assembly area and find the other end of the line; it will be secured by a small nut or a screw. Loosen that and pull the thermocouple out, take it to the supply house to get the new one, and replace it with the new one.
3. Make sure the thermocouple is in the pilot flame when you light it (lighting requires holding the knob or button down, let the flame burn a minute or so before you let up).
4. When the pilot is staying lit and the thermocouple is properly positioned, make sure you turn the knob to the "ON" position and close the access door.
Like I used to tell my plumbing class, "There's nothing to it, if you know how to do it".