Below is a walkthrough for A Pet’s Rest side quest in Tiny Tina’s Wonderlands. To obtain this quest, speak with Lisha in the Overworld after completing The Godswell area.
Reach cave
Head to the objective in the north and enter the dungeon. Complete 2 encounters then loot the chest that spawns at the end. Exit the dungeon via the portal.
Talk to Lisha
Return to Lisha and speak with her to complete the quest.