This quest will be initiated once you reach Chapter 3. It’s more of a passive quest in that it will update as you go along but it’s central to the main story of the game.
At the beginning of The Witcher, Geralt returns to Kaer Morhen without any memory of what happened to him. Throughout the chapters Geralt will talk to various people from his past that will help him to remember who he was.
Speaking with Triss Merigold at the beginning of Chapter 3 will help Geralt uncover who he really is. She will refuse to say what he was really like and instead insists that Geralt find his own path.
You may want to come back to Triss for more insights throughout the game. You can also speak with Thaler, Dandelion and The Lady of the Lake in Chapter 4. The choices you make along the way, such as choosing Triss or Shani, the Scoia’tael or the Order, killing or sparing Vincent (the werewolf) will update this quest.
This quest will not be complete at the end of the game so you’ll have to play Witcher 2 to find out what happens to Geralt next.