If you are having trouble with this quest do as I advise:

Get stopwatch and stand near gryphon outside Velia.
When it arrives start stopwatch and go to npc Croix.
Around 3:30 on your stopwatch start quest timer and get on the gryphon's chest.
Flight should start when your stopwatch will show 4:05 and it takes around 4.5 min to travel to Iliya.
You should have around 40 seconds of quest timer left when you arrive. NPC Diega is close to the landing site.

You can use a map to teleport to Iliya Island in case you messed up the timing with the griffon.

Careful, the timer starts as soon as you open the [ready] dialog, even if you didn't confim yet.

I made a video of the quest here https://www.youtube.com/watch?v=mSXYn-flCPA