Go to Sauromugue Champaign (L-10) (must go up the cliff at (L-8) first), 3 Sabertooth Tigers wait outside the cave. You can ride a chocobo into the cave to avoid them, but bring along an Instant Warp Scroll if you do not want to fight them on the way out.
Wait for the Old Sabertooth to die naturally (~5 minutes). He will not aggro you if you stay at the back of the cave, even with the battle music activated. If you accidentally kill him, wait five minutes, then re-examine and he will pop again.
Examine the bones and bring the Old Tiger's Fang back to Perih to complete the quest.
If someone else started the process prior to you, actually seeing the old Sabertooth die (see the death notification text) enables you to get the key item without first examining the bones to spawn the Sabertooth.
Note: If you do this quest with several of your characters, make sure you aren't holding the Ranger's Necklace from another character, as this will prevent you from completing the quest. If you do, you will get a message that you should return after sorting your inventory (normally refers to a full inventory).
In order to become a truly great hunter, one must be accepted by Mother Nature. In order for that to happen, one must gain the fang of a tiger, the fanged king. But it does not sound like any old fang will do...