Slap's in KCK has the best pork ribs I've ever had. Joe's in Mission KS if you're into BBQ sandwiches (z-man is legit). Jack's Stack if you want great burnt ends and really good sides (they're all over, but Crossroads is probably the closest to the airport).
If you're in Westport, an underrated BBQ place is Char Bar. It was incredible and one of my favorite places in KC.
If you're further south in KCMO, BBs Lawnside is really good too.
I personally pass on Q39. It was ok and overpriced, didn't understand the hype. Tried it a second time and still didn't get it.
Whatever you do, know that KC burnt ends are usually very good, but their standard brisket is not (too lean).