Unlikely Valentine

Like Ellie said, Nick should have gone to Park Street Station to the east so head there and enter the station.

Fight your way through several Triggermen in the entrance. Continue below to the actual station and engage more Triggermen. Continue following the tracks and kill the last few Triggermen to find the entrance to Vault 114. Interact with the vault door controls to open the vault and gain access inside.

The main door requires a key to open. The alternate entrance to the vault is to the NE room. Go downstairs and follow the quest marker, fighting off more Triggermen along the way. Once you’ve reached the next part of the vault in the large cavern, enter it and drop down below. Continue to the corridor and you’ll find the sliding door leading to the actual Vault 114.

Continue forth to the next sliding door and you’ll hear a guy named Dino and Valentine talking. Dino will immediately turn hostile upon spotting you so kill him and loot the Overseer Door Password from his body and use it on the terminal to open the door and free Valentine.

Before leaving the room, don’t forget to get the Speech Bobblehead from the table, as well as the V114 Interview Holotapes (#03, #21, and #87). Follow Valentine to the lower levels and defeat the Triggermen you’ll encounter along the way. You can also check out the different rooms along the way.

Continue following him past the bathrooms and to the stairs. Before reaching the sleeping quarters, look for an issue of Astoundingly Awesome Tales on a small container beside a barricade of Vault-Tec containers before the doorway.

Continue forth until you reach the nursery area. Kill the next batch of Triggermen and continue following Valentine until you reach the vault entrance. You can collect Vault 114 jumpsuits before leaving as well. In the exit, Skinny Malone and his boys will confront you. You will have several Charisma speech checks to trigger here so make sure to quicksave. If you managed to convince Skinny Malone to let you walk free, follow Nick to the exit. Upon exiting, Vault 114 will be permanently marked in the map and the quest will be complete as well. He’ll ask you to visit him in his office in Diamond City. This will start the next quest as well.