To trigger this quest you need to reach the eleventh day of Spring. I am not currently positive whether this is the exact criteria, so feel free to correct me in the comments if you know better. Regardless, you’ll receive a letter in your mailbox when the quest is ready for you.

Robin has lost her Axe, and if you check the description it says she lost it in the woods south of Marnie’s Ranch. Take the following steps to retrieve the Axe:
  • Head south of your farm to Marnie’s Ranch. You can get there directly from your farm, if you’re willing to chop through the overgrowth in the south. You can also go through Pelican Town and loop back west.
  • Walk south until you hit water, then follow the water south until you find a bridge leading east.
  • Cross the bridge to find a sparse forest beyond. Once you’re in the trees, head east to find a cliff ledge, and follow the ledge south until you find the Lost Axe
  • Take the Axe to Robin at the Carpenter’s Shop, north of Pelican Town. She’s found there ‘most days’ (on Thursdays, at the very least) from 9 am to 5 pm, behind the front counter. Gift it to her like you would a present to any NPC.
Bringing the Lost Axe back to Robin will complete the quest. Check the Journal entry for Robin’s Lost Axe for your 250 gold reward.