It's probably better to think of it as a percentage of their range because this is how often they will 'have it' during play. Don't forget that you're also way behind pocket pairs 66+.
Say that the opponent's range is something like all pairs, all broadway, all suited aces, and all suited connectors. If you had A5s say, and accounting for card removal effects, villain has 230 starting hands in his range, 31 of those contain a queen, and 46 more give him a better pocket pair. So if he has this range then he has the Q 31/230 = 13.5% of the time, and he has a better hand (31 + 46)/230 = 34% of the time.
In this situation I'd probably look to get max two bets in postflop. When he wants to get all-in, eg he raises the turn or jams the river, he almost certainly has the Q. Against a TAG with a worse hand, I'd expect them to bet once and give up or maybe call once postflop.
Paired boards are not as safe as many people think, especially if the paired card is a high card. For example, ignoring card removal effects, if the flop is not paired, there are 120 hand combinations that give villain top pair (3 cards for the pair * 10 possible ranks for the kicker * 4 cards of each rank for the kicker), but if the top card is paired, there are still 88 hand combinations that yield trips (2 cards for the trips * 11 kicker ranks * 4 kickers of each rank). That's not a huge reduction in the number of combinations.
Compare that to the number of possible flush draws on a two tone board. There's only 55 flush draws (11 choose 2). So, everything else being equal, you're more likely to see trips on a paired flop than you are to see a flush draw on a two tone flop. Just to put it in perspective.