I am doing an essay on how Romeo and Juliet mature throughout the play. I am supposed to identify the exact moment that each transitions from child to adult. I have no problems finding examples of maturing, but I'm not sure which is the "exact" moment for each.

For Juliet, her moment of transition comes when she cuts herself off from the Nurse in act three, scene five. After Romeo's banishment, the Nurse tells Juliet she should just ignore her union with Romeo and go ahead in marrying Paris, since Romeo is out of commission anyway and her parents will approve of Paris.
Juliet views this as a betrayal—a hypocritical one since the Nurse previously supported the union. The Nurse is the closest thing she has to a mother-figure. Once she realizes the Nurse is no longer supportive of her, she decides to go it alone:

Ancient damnation! O most wicked fiend!Is it more sin to wish me thus forsworn,Or to dispraise my lord with that same tongueWhich she hath prais’d him with above compareSo many thousand times? Go, counsellor,Thou and my bosom henceforth shall be twain.

This is Juliet declaring her independence from the adults in her life. She is hereby estranged from her parents and her Nurse.
For Romeo, the moment is more elusive, since he remains extremely impulsive throughout the play. When he is banished from Verona, Friar Lawrence chides Romeo from crying like a child (calling his tears "womanish"). When he hears Juliet has died, he immediately sets his heart upon suicide so he can be with her. However, one could argue he starts showing true maturity in act three, scene five as well.
After his wedding night with Juliet, it is Romeo who insists he must leave before daylight breaks. When Juliet asks if they will ever meet again, he comforts her by saying they will one day laugh about their current troubles. He is trying to be strong and optimistic for Juliet's sake, and this is a mature response.
https://www.playshakespeare.com/romeo-and-juliet/scenes/325-act-iii-scene-5