Hello there @Mikaza98 The requirements for this quest changed in a recent update. To start the Eon Duo Quest, you require all non-legendary caught data for Hoenn and Hoenn Champion. It appears you are still missing the caught data for several Hoenn Pokemon (Spinda, Altaria, Zangoose, Chimecho, Gorebyss and Salamence). Once you have the caught data for these Pokemon, you will be able to progress with the quest.
If you have any questions, please let me know.
Houndstone