I'd try finding some breeders and rescues in your area to see what they think. If quakers are not common where you are, and there is a reasonable demand for quakers (from people who will provide good homes), then responsible breeding can be a good thing.
As far as keeping a tame pair, it's hard to predict. Birds that are intended for breeding are usually parent-reared, or if handfed, are kept with others of their species and not handled by people after weaning. This is done so they identify more with other birds and can bond more easily with a mate. They often aren't tame because of this. Very tame, human-bonded birds may not necessarily be interested in breeding at all. Or a tame bird may bond with a mate and lose interest in people.
This isn't always the case - some breeding pairs remain completely tame, or will be territorial when breeding and tame the rest of the year.
It depends partly on how the species breeds in the wild - in many parrot species, nest sites are very rare and something they must fight over. Birds that get along well outside of breeding season must defend their nest sites fiercely when breeding. In others species, like many conures, pairs in a flock will share a nest, so they're much more likely to remain tame and friendly with trusted people when breeding. Of course, quakers are unique among parrots in that they build communal nests, but each pair has its own 'apartment' that it defends, and I'm not sure how they would behave.
Sadly, there aren't really any genetic tests that are done with birds. Often, it's the 'problem birds' - ones that don't make good pets for whatever reason - that end up being bred, because people are afraid of a beloved tame bird losing tameness if it's bred. But this is unfortunate because it means people tend to be breeding for undesirable traits. It is good to try to breed from birds that don't pluck, don't have any heritable health problems, and don't show unusual aggression or nervousness, but other than that, it seems there is little you can do besides provide the best care possible.
