When do you unlock index in persona 3?

You unlock the Index in Persona 3 after you complete the Tartarus: 4th Block portion of the game. This is around mid-July in the game's calendar.

Once you reach the 4th Block, you'll encounter a new social link character named Fuuka Yamagishi. By progressing her Social Link, you will eventually be able to access the Index, a database of information about the Shadows, Personas, and other aspects of the game's world.

Hope this helps!