Harlow will tell you that the golden horse bolted to the north. By keeping to the east side and riding for about a minute or so, you will detect a golden horse from afar.
In case you don’t find it easily, here are the exact coordinates, (-1757, 3336, 0241) to the location.
As you spot the golden horse, try to sneak up on it so you don’t spook it again.
Make sure your stamina bar is full to tame it. In case you don’t have a full stamina vessel; you may need to quickly consume a stamina-restoring potion.
After making your way back to Snowfield Stable, speak to Penn and Harlow for the last time.
They will be so grateful that you were able to find Zelda’s Golden Horse that they will reward you by allowing you to keep it.
Upon successful completion of your quest, Harlow will also reward you with a prestigious Royal Bridle and Royal Saddle, revitalizing Energizing Elixir, and a generous sum of 50 Purple Rupees.