Speak with Aldous, in the Duke's Audience Chamber in Gran Soren Castle, to start this quest. Tell him you want to aid the corps on a hunting expedition. He wants you to help make the roads safe by killing a Griffin that is attacking some caravans.

This begins the quest:

Griffin's Bane
Speak with the volunteers

Objective
Royal Order Quest
A Griffin is stalking caravans. Join the volunteers who plan to vanquish the beast

Details
Talk to the captain of the company of "elites" to learn of their strategy. They await you outside the southern gate.

Exit Gran Soren through the Southwest gate and talk to Ser Georg standing down by the road. He tells you they will circle the capital, striking upon it if it comes along their way. The strategy is to attack it when it sets down. Join the campaign.

Follow the soldiers as they patrol the fields. Soon you will reach an open area, and a short cinematic will show a marked zone on the map containing some bones. Scout the area for a goblin, then kill it and put it in the marked area to be used as bait to lure the Griffin. Alternatively, if you cannot locate a goblin, try a spider. Soon the Griffin appears. Attack it. Try and aim at the sides (the wings), especially if you are a melee character. Eventually, after you wear the Griffin down enough, it flies away and escapes. Ser Georg thinks it is heading to the Bluemoon tower, and instructs to give chase. Pursue the fleeing Griffin.

To get to the Bluemoon Tower, head northeast from Gran Soren past Windbluff Tower, the take the road east/southeast through the mountains. When you reach Blue Moon Tower, search for the Griffin and battle it using the strategies mentioned previously. The Griffin might escape to the other side of the tower, which will require you to climb to the tower rooftop. If you previously helped Steffen in the quest "A Troublesome Tome", he shows up in the middle of the fight, to lend you his aid using arcane magic.

This tower has lots of good loot to be found scattered around, so search thoroughly after slaying the Griffin before returning to Aldous to report your victory.

This completes the quest, Griffin's Bane.