Could be your B&G doesn't like your Quaker. This isn't unusual when parrots are jealous of one another. Personally, I don't think it's a big deal that they don't get along. With birds that different in size, they shouldn't be allowed to interact together, even with supervision. Just my opinion.

As for the runny nose, how do you know it's allergies? Did an avian vet advise you of that? If you haven't been to the vet for it, then I strongly suggest you do so.
Cute photos