You learn that Zelda's beloved golden horse behaved strangely and ran away when the princess approached.
Quest Location
After you start the side adventure Potential Princess Sightings!, you can talk to Penn at -1641, 2583, 0233 to begin this quest.
Find the Missing Horse
Before you go, you might want to either upgrade your stamina wheel by getting at least four Lights of Blessing from Shrines and praying to any goddess statue, or by getting an Energizing Elixir, or resting in the Malanya Bed if you have unlocked that with Pony Points, etc.
If you are ready, look north from the stable, where Harlow said that the golden horse ran off, and you will see a Frost Gleeok (a three-headed dragon) flying around. You don't need to battle the Gleeok to find the horse, so just steer clear of it and go north.
Eventually you will find a herd of horses, including Zelda's golden horse. You will need to sneak up on the golden horse and mount it, then ride it back to the stable.
Note also that there is a geoglyph here.
When you mount the golden horse, you will need to repeatedly tap the soothe button. If you haven't upgraded your stamina wheel, you might not have enough stamina to soothe the horse enough. Once the horse is soothed, you can try to tame the horse by pressing the indicated button to soothe it each time it obeys you, such as after you jump over an obstacle, or after you spur it on with a speed boost, or after it calms down from being disobedient.
Ride the golden horse back to the stable and talk to Harlow while still on horseback to complete the quest.
Quest Reward
You get to keep Zelda's golden horse, and Harlow gives you a Royal Bridle and Royal Saddle, and an Energizing Elixir.