why can't you say she answers right away TO the question? there's alla before the word domanda.
Because one "answers a question" rather than "answering to the question". One answers TO the person asking the question, rather than the question itself.
I think there's a slight difference in meaning: "subito", as far as I understand, is pretty much "instantly". "Quickly" has a slightly greater range of meaning, and it might even mean that she talked quickly while delivering the answer rather than starting to respond at once. (Native speakers, feel free to correct me...)
You can always try reporting it. If the big green owl agrees that "quickly" is OK, it will be added.