From Thundermist Village Station head to the griffin fields and left towards the hill where Bloodtalon spawns.
Just as you step off the road there is a large dirt area with lions and griffins ( 1227, 0, 594 ) /waypoint 1227, 0, 594. Use the tab key to quickly search until you find griffin eggs.
Note: There are more places where the eggs spawn, but this area is a guaranteed spawn. (Other place like at ( 1000, -2, 175 ) /waypoint 1000, -2, 175).
Take the egg to a griffin tamer while it is still fresh. You have 7 minutes 20 seconds to complete this step or the egg will spoil.
Hail the griffin tamer to complete the quest and receive your reward
This quest is now optional after major travel updates in 2010. It now only rewards experience. The old reward (below) is retained here for historical purposes.