Is there a specific trick you have to do to get Kilgore's quest?

I have beaten Gargos at least once. I’ve gotten Eyedol’s, but never Kilgore’s. What’s up with that? Do I need a UltraTech member in my team?

Yes you need an UT member I believe.there is an old thread with all the specifics somewhere in here. Use the search tool to find it.