There are 4 moonstones at the following locations.

At the start of the game during the theater mini-game, score 100 nobles impressed and also impress Queen Brahne. Continue until you take control of Steiner for the first time then speak to Queen Brahne.

There are 4 colored stones on pillars on the Conde Petie Mountain Path. Collect and insert all of them in the statue past the Moogle.

Give an emerald to the friendly Jabberwock in a forest on the Forgotten Continent.

Complete the Stiltzkin side quest and purchase his last bundle for 5,555 Gil.