Unlikely Valentine

Unlikely Valentine is a main quest and achievement/trophy in Fallout 4.

Quick walkthrough
Upon following the advice of Diamond City locals, the Sole Survivor sets out to the detective agency, only to find Nick Valentine is missing. Ellie Perkins gives the Sole Survivor the quest.

Detailed walkthrough
After speaking to Ellie Perkins head out of the main gates in Diamond City. Follow the Pip-Boy to take you to Park Street Station. Fight your way through the gang members until you reach Vault 114. Upon entering the Vault you will come across 3-4 Triggermen armed with sub machine guns. Kill them and move on. Carefully fight your way through the vault until you come to a large open Atrium. Head to the left and climb the stairs. In front of you there will be Dino. Dispatch him and use the control panel behind you to open the door to the Overseer's office. You can also sneak in and listen to a conversation between Valentine and Dino: Valentine convinces Dino that a crime boss is upset at him about his gambling habit, Dino then runs out of the room to square things off with said crime boss leaving the coast clear to free Valentine.

Speak with Nick (be sure to grab the Speech bobblehead behind him on the table). After talking with Nick he will lead you out of the vault. After a few fights with Triggermen, you will be stopped by Skinny Malone and Darla. Depending on your Charisma level and how you deal with Malone and Darla, it is possible to turn them against each other or leave the vault without fighting at all. If you had Mama Murphy use The Sight before coming to the vault, you will receive an additional speech option that will allow you to go free without any violence.

After exiting the vault, follow and aid Nick into getting back to Valentine's Detective Agency. You will be engaged by Raiders and mongrel dogs while returning to Diamond City. Alternatively, you can fast travel to Diamond City and find Nick already in his office.

When you arrive speak to Ellie again to claim your rewards.