Return the pup to New Combine Foothold. If you end up in combat, be careful to position the mob facing away from the pup. The pup can receive frontals and it will die. If it dies, you have to delete the quest and start over again.
If a player is having difficulty navigating through the forest with the NPC, it is possible to complete the dialog to the point that the Accept Quest panel is displayed, and then run or fly to the camp with the dialog open, without clicking accept. Once safely in the camp, clicking the accept button will spawn the NPC next to the player and the quest can be turned in.