Life as a servant robot might have its perks, but it's hard to blame any robot for wanting more. That is the case with one of your companions, Curie. To trigger her special quest, first you must find her. To do so, complete the Vault 81 and Hole in the Wall side quest.
Recruit her as a partner and build a bond by keeping her as a companion and engaging in activities that she approves of. Consult our Companions Guide to speed up the process. When she trusts you enough, she'll share her wish to become more human.
This is one of those rare quests that doesn't involve confronting hostiles. Simply travel to Goodneighbor and visit Doctor Amari at the Memory Den. She even has a Synth all ready with a body to donate for this quest. Proceed with the operation and come back after 24 hours.
When you return, you'll find the Synth's caretaker and Curie. Engage in the resulting conversations however you like to complete the quest.