The Kyurem quest is the quest done in order to capture Kyurem.
Kyurem is the only Tao Trio member currently catchable. It is usable in Ranked PvP, despite having more than 600 base stats.
Before starting the quest, make sure you have the following:
Make your way to the northwest of Evergreen Waterway to reach Galeras Island South. Once there, go to the northwest once more to reach Galeras Island West. Next, head to the middle of the island, Reshiram will be waiting there.
After defeating both Reshiram and Zekrom, head back to Eumi Island and speak with the aforementioned old lady. She will tell you about the origin story of the Tao Trio, when suddenly you hear someone crying.
Go to Eumi Island House 1 and speak with Joey's Mom. Joey's Mom explains that her son has gone missing and requests your help. The Grandma then appears and gives you a mysterious potion. Head back to Evergreen Waterway and surf to the northeast to reach Enyor Island Forest. From there, head to the northeast until you see a cave. Go into, and through, the cave. There are multiple correct paths in the cave and Cryogonal will be blocking parts, but they will disappear once defeated.
Outside the cave, in Enyor Island Clearing, your non-Dark type Pokémon will complain about headaches. To pass this part you will need at least 3 Dark type Pokémon in your party. Then make your way through the clearing, you will find Joey in the middle of a small lake. But before talking to Joey, you will have to fight an Alakazam and a Weavile. Speaking with Joey will start a battle with Kyurem. This is not the Kyurem you will catch, so you will need to defeat it to continue. Once Kyurem has been defeated, you will be teleported back to Eumi Island House 1, where Joey's Mom will reward you with a Reroll Ticket (Note: You will only get the Reroll Ticket if you beat Kyurem every time).
Head back to Enyor Island Forest and find Kyurem. It will be at the northeast, in front of the cave entrance. You will need to defeat this one as well. Once more, you will need to find Kyurem and fight it. It will be on the northwest of Enyor Island Forest but this one is the catchable one. Prepare your sync Pokémon and make sure you have enough Pokéballs or a Master Ball to avoid any mishaps.