Not really a quest line but a trigger event.
It happens more or less automagically - after you bring her Aiko - on a random day (happened for me on day 13) on Street K, Downtown. If you go choose to go back and tell Lily, then the deal is rescinded and you won't have to bring her anyone any more. You can if you want, but you don't have to.
If you keep silent, the game will progress normally.
In v0.36.3 the event fired erroneously* but according to
Daedolon, that got fixed since.
* It would fire regardless of where you are. Happened in Utako's house for me. I had to forcibly drag the idiot back to the church.