KK and QQ would probably have been PF raised, so those are out of his range. And although his flop bet was 3/4 pot, to his stack this is pretty weak, and most likely either a feeler or steal. I still don't think he has a strong hand yet.
The Q hits the turn and he (traps) check/raises you all-in.
me that out of his range he is probably holding Qh/xh, Q9, Q7. I've discounted the straight draws, because he said he doesn't have them. Normally, I wouldn't do that, but, he said that he would show. People will generally lie about what is in their hand, but, if they say they will show they do it.
Either way you are ahead of most of his range here, so a call should be pretty easy.