Hunters Guild Leader Arcturus:
Fill 10 Hunter Guild's Bestiaries. You can fill them by defeating 10 of the monsters illustrated in the Bestiary. You can select even those you have defetead before.
Note that it is useless to kill monsters of other factions.
Click Menu > Bestiary, and the Bestiary page will open. Select the Hunter Guild as your faction and defeat 10 of the listed monsters to complete the mission.