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Bet. He doesn't have jacks, he either has kings or queens, with the Q out its probably slightly more likely he has kings.

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6 ways to have Kings, 3 ways to have Queens.

You'll pay 2 when he has Queens, and you'll get 1 when he has Kings.

So if the only hands he can have are Kings or Queens, it really doesn't matter if you bet or check. However, considering there are some other hands he could have (JJ, TT, 99), you should probably just check. None of these "other" hands are very likely, but even if he has them 1% of the time, it turns a river bet into the red.

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I'm pretty damned sure he doesn't have JJ, TT, or 99.

Since you have to call a raise, you have to be good here 66% of the time. With the weighed hands being KK (6) and QQ (3), I think that's exactly what you need: a ratio of 2:1 that he'll call with rather than raise. There's even a small/slim possibility that he doesn't raise QQ on the river, but I doubt it.

Rob