Triss will give you this quest when you talk to her in her house at the start of Chapter III.
Triss will tell you that she can't restore your memory, but that by taking stands and making decisions, you might discover for yourself who you really are. As far as we're aware, this quest won't lead to anything or award you with anything -- or even end -- but you'll update it from time to time during the course of the campaign.
Here are some of the places where you can update the quest:
Thaler at the New Narakort Inn (Chapter III). During the quest
A Posh Reception
, when you ask Thaler who he is exactly, he'll in turn ask you if your reasons for hunting the Salamandra are personal or ideological. If you answer "ideological," then you'll realize "I'm driven by the fact that I'm a witcher." If you answer "personal," then you'll realize "this has become personal for me." If you answer "both," then you'll realize "my motivations are complex."
The werewolf in the
(Chapter III). During the quest
Lock and Key
, you'll encounter a werewolf, who will explain that it only hunts "the weeds of society." If you let the werewolf go (so you can perhaps cure it later), then you'll realize "not every monster deserves to die." If you kill it, then you'll realize "a monster is a threat that needs to be annihilated."
Dandelion in the country inn (Chapter IV). While talking to Dandelion about what happened in Vizima after you left, if you say "I only kill humans in self-defense," then you'll realize "I don't like killing and I resort to it only when necessary." If you say "Salamandra and their kind deserve to die," then you'll realize "I have no objections to killing humans if I believe they deserve to die." If you say "I have nothing against slaying humans," then you'll realize "I have no objections to killing humans."
The Lady of the Lake on
Black Tern Island
(Chapter IV). When you ask her if she can help you regain your memory, she'll ask you if you believe in destiny. If you answer "I do," then you'll realize "I tread the path destiny traces before me, and I have a mission to complete." If you answer "I am master of my own fate," then you'll realize "I don't believe in destiny and [...] I'm responsible for both my successes and failures." If you answer "I believe in something more," then you'll realize "I cannot say what influence destiny has on my life."
King Foltest in the
(Chapter V). He'll ask you about the woman you chose in Chapter III. If you say, "Shani / Triss and I, we're close," then you'll realize "Shani / Triss is the love of my life." If you say, "We're lovers, nothing more," then you'll realize "though I feel good with Shani / Triss, I'm not ready to love her." If you say, "Shani / Triss is a friend," then you'll realize "Shani / Triss is first and foremost my friend."