After reaching the Elven Kingdom, talk to Stan to start this quest. Set him as your buddy then head to their house. Enter the room beside the Stan's to fight Shelly. She has a lot of HP but that can be easily lowered with the help of stacked status ailments. Make sure to block her attacks though since she hits hard.
After defeating her, the quest will be complete and Stan will become your friend. Don't forget to go back to the house and talk to Shelly to add her as a friend as well.