This pivotal moment occurs in the book *New Moon* when Bella Swan travels to Italy to find Edward after he leaves her. She seeks him out in Volterra, the ancestral home of the Volturi, a powerful vampire coven.
While Rome is mentioned in the story, it's not where Edward reveals himself. He chooses to do so in Volterra because he wants to confront his family and potentially face the consequences of breaking the Volturi's laws.