Details: Should you go to the Thrifty Megamart within Royal Avenue after defeating the Pokemon League, you can speak to a veteran in the store that will ask for your ask. You see, thirty years ago, he idolised and always tried to beat a collection of trainers who trainer certain Eevee Evolutions. Since he cannot do it for himself as he now has a wife and child, he will ask for you to travel across The Alola Region in order to battle these given trainers.
Eevee Evolution User Locations
Vaporeon User Polly - Trainer School (Route 1)
Jolteon User Jane - Malie City
Flareon User Chad - Tidal Song Hotel (Heahea City)
Espeon User Ishaan - Blush Mountain
Umbreon User Braiden - Hau'oli Cemetery
Leafeon User Linnea - Hano Beach
Glaceon User Rea - Iki Town
Sylveon User Sakura - Seafolk Village
After defeating all of these trainers, you can return to the veteran in Thrifty Megamart. However, your Eevee Quest is yet to be over, as he himself will be the Eevee User. Defeat him to receive Eevium Z.