This quest will start while talking to Mama Murphy shortly after completing When Freedom Calls. Diamond City is a long walk to the south and there's really no immediate need to go there. Upon reaching the gate, you'll find Piper. Talk to her to trigger a cutscene.
After the gate opens, follow her inside. Have a conversation with McDonough as well. After the cutscene, Piper's quest “Story of the Century“ will become available. Before doing so, spend some time exploring the city.
Your next objective to continue with the story is to talk with Nick Valentine. Head there by following the quest marker and talk to Ellie to learn that Nick has been missing while tracking a gang for the latest case he’s working on. Offer your help to complete the quest. The next quest will start immediately afterward.