Just like Cait, Piper is a little easier than you'd expect. Mainly because there isn't really a quest in which she triggers.
There are two tactics you can use. The internet's long process, or my spoiler heavy version. The internet will tell you to be nice and unselfish on your travels while using you. Resolving issues in a peaceful manner to get her to talk about her little sister choose the option of family and after a while on the road, she'll have thought about it and then decided you were right. You can flirt with her if you want to try a romance path, which can be done as Male or Female. Eventually the last converstaion will spawn in which you can romance her.
My version,do a bad boy/girl playthrough with Strong and Cait. Their perks are the most useful in the end game quests. When you come to the end of a faction (Minutemen, Brotherhood of Steel or Railroad) with the "Nuclear Option" quest. For Piper her admiration of you increases much more severely if you choose Minutemen or Railroad. Once the institute is destoryed, take Cait or Strong back to wherever you left piper and she'll trigger a conversation about the institute being destroyed. This should cut out most of the wait to the next conversation about her little sister. My game triggered both conversations before I asked her to follow me.
You'll still have to use her normally to get to the romance conversation, but this is pretty much a little quicker than the normal route. You now have the perk "Gift of the Gab", this gives you double XP for for speech challenges and discovering new locations.