It is worth noting that this can be summarized by Phil's own suggestion:

One approach is to add up the bits of information each expert gives, with positive bits for indications that Q(O) and negative bits that not(Q(O)).

That is, you can interpret the log of the odds ratio as the evidence/information that A gives you beyond Q. Adding the evidence from A and B gives your aggregate evidence, which you add to the log odds of the Q prior to get your log odds posterior.