How can I get the Giglamesh Gene?

You need to collect six yellow gifts from golden chests in the last three dungeons. The golden chests are unlocked by completing the Secret quest chain after Chapter 4..

It requires those 6 yellows gems from :

I Fancy Final Feast-Edge of the World Box/Dungeon

II Darts- Edge of the World Box/Dungeon

III Memory Drop IV- Primodia Box/Dungeon

IV Moonlight Shadow- Primodia Box/Dungeon

V Rakshasa Lutos- Theia Box/Dungeon

VI Iris Shield- Theia Box/Dungeon