They can smell the aggitation and fear of other animals and their own experience has taught them it can be a scary place. Dogs will react to fear with either flight or fight, and it may be different for any certain circumstance. I always try to segregate my dog away from other pets while at the vet especially at times when it's busy and avoiding very close proximity to another animal may not even be possible. It's especially difficult to keep an eye on your dog while you're trying to write a check or discuss instructions with the vet or the person at the desk, etc.
There was a mistake made here by the women putting her dog down right in front of yours... a strange dog invaded the air space of your dog and in an environment that is already nervewracking for most dogs. But of course I understand how you feel... it is mortifying when our dogs do something that makes perfect sense for a dog though isn't acceptable in the human world. Try not to feel bad, you were the one put in the awkward position of having a strange dog invade the space of yours without your permission and when you weren't in a position to completely have eyes on and control of your dog since you were dealing with the checking out bit at the desk.
Having a big dog, I just feel better if he's muzzled at the vet so I don't even have to be bothered by worrying about what other people and their dogs are doing. I makes me much more relaxed which in turn relaxes him. Since Yoshi is a pup he pretty much loves everyone and would likely be submissive to other dogs, but now that he's getting older I'll likely use the cage muzzle with him at the vet not so much because of him, but because of ME... I just feel better not having to worry about anything going wrong regardless of who made a mistake or why.
Really, try not to feel so bad. No dog can be expected to like all other dogs nor accept them very suddenly being forced upon their air space. Dogs NEED to be able to check out another dog from a distance first so they can gauge for themselves what the other dog is telling them about themselves and decide whether or not they WANT that dog in their air space or not. This dog was forceably thrust upon yours without warning and without being given an opportunity to check out that other dog from outside its air space first, so Quincey's normal reaction was to protect both himself and you from sudden strange dog invasion. Had the woman put the puppy on the floor outside of your dog's air space and the two of them could have a chance to check each other out and read each other's body communication they likely would have been just fine with each other after doing proper dog ritual greetings.
Since the bite was light (it doesn't take much force at ALL to break the skin) this could have been Quincey correcting the bad behavior of this youngster for committing the enormous sin of instantly invading his air space. Dogs bite to correct especially youngsters... it's the force of the bite and the reason for it that matters, and it could be that this was not at all aggressive behavior but normal correction behavior that required immediate action. This dog was a puppy, so automatically Quincey knew that his own rank was higher than the puppy's, and this child-dog needed correcting (to all dogs a puppy is of lower rank... that's why puppies always behave as submissive and eager to please to everyone).
I'm so sorry this happened. From your description of the incident it wasn't your fault or Quincey's fault. He behaved normally for a dog and neither of you was given an opportunity to head off what happened. You were pretty much stuck with only being able to correct Quincey after the fact. But I know exactly how you feel. I always feel mortified if my dog does something inappropriate in the human world even though he was defending himself or me because of some bozo or some other bozo's badly behaving dog thrusting themselves upon us.
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