The Professor's Protege - Griffon (Coernix Station)
First Available: Chapter 7
Other Requirements: Complete The Professor's Protege - Myrlwood Firefly
Location: Coernix Station
Rewards: 12,000 EXP & Safety Bit
In order to find Sania at the location you see her in the screen shot above as well as accept this quest you'll need to complete all of the previous 'The Professor's Protege' quests from her. For this quest you will need to head over to Lestallum and speak with one of the Diner owners here (the one in the market - marked on your map) who will offer you a Hunt for a Griffon.
The hunt is called 'Ruler of the Brave Skies' and as far as I can tell there are no requirements for accepting the Hunt. Much like all of the other quests that require you to collect an item from an enemy on a Hunt, there is only a chance that you get the item you're searching for.
What this means is you may have to complete this hunt numerous times to get the Griffon Feather that you're after. You can increase your odds of getting this item by having Ignis make Mother and Child Rice Bowls (Oyakodon) at camp which increases the drop rate of items for you.
Alternatively you can also save before the battle and keep resetting until you get the item you're after. Once you've obtained the Griffon Feather return to Sania and turn your quest in.