You have to start it during the quest, but I don't remember if you have to finish it during the quest as well. All I know is that if you don't start it, you can't have it.
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I was curious, so checked the code. you're right, you do have to start it while doing that quest. However, you only have to ask about faries, nothing more. The flirting can happen later, asking about fairies starts it out and that requires the mainquest, but the rest is independent of that. Everything requiring reputation can happen at any point after that, so just make sure to ask her about fairies when first tasked with finding a fairy.