After successfully infiltrating The Institute, talk to Elder Maxson to receive this quest. He'll ask you to find Dr. Madison Li and convince her to return to the Brotherhood. You can convince her through a series of Charisma speech checks with increasing difficulty. You can quicksave then reload if your persuasion attempt failed. If you can't convince her, you have to explore the FEV lab inside the Institute.
The entrance to the FEV lab is either through the Master terminal or a Novice locked door in the back pathways of the BioSciences Division. Locate the holotape Brian Virgil personal log 0176. Deliver the holotape to her to convince her successfully. Once done, report back to Elder Maxson.