You can complete this little quest after the party is attacked by Ba'Gamnan in the Lhusu Mines. The first step to completing this quest is to finish the "Cry of its Power" hunt petitioned by the moogle Pilika (found in Bhujerba's Kjus Skygrounds). Once that is taken care of talk to Pilika again after receiving your bounty. She asks you to retrieve her diary from the second floor of the Technicks shop. You'll then get a Merchant's Armband that must be worn to access the second floor.

Go to the Technick's shop and go to the second floor. You should find PIlika's Diary in the large bookcase on the wall. When you pick it up you have the opportunity to read it. Next it's time to return the diary to Pilika.

When you give the moogle her diary back she asks if you read it or not. If you answer "Read it" Pilika will reward you with a Kilmweave Shirt. If you say, "I've done nothing" then you get a Shepherd's Bolero. It doesn't matter if you answer truthfully or not - the reward is based solely on your answer.