This page explains how to start and complete the Hunt for the Giant Horse side quest.
How to Start the Quest
Talk to Straia at the Mounted Archery Camp, which is west of the Highland Stable, which can be found southeast of Lake Tower, near Ka'o Makagh Shrine.
How to Complete the Quest
Go to Owa Daim Shrine on the Great Plateau, and float southwest from there to Taobab Grassland. The giant horse is walking around with the normal-sized horses. Mount the giant horse and soothe it until it lets you ride, then ride it to the Mounted Archery Camp and talk to Straia while on horseback to complete the quest. You will have to get past two Lynels, and a group of Bokoblins on horseback.
Quest Reward
Straia gives you a Silver Rupee. You can register the Giant Horse at a stable, so be sure to do so.