Green is a female trainer from the Kanto region. Initially being designed as a random NPC for Ken Sugimori artwork, she really got established as a character in the "Pokémon Adventures" manga. The female protagonist of FireRed and LeafGreen, Leaf_Game, also is based off her design.
Game canon
Green made her first game appearance in the alternate universe Let's Go Pikachu and Eevee, being a girl who wants to capture Mewtwo. She seemingly goes crazy once she finds out the player (
Chase or
Elaine) caught it, and tries to capture them too, but she seems to be good friends with Red_Anime and
Blue Oak.
Pokémon Evolutions
Green is a trainer trying to complete the Pokédex, and Mewtwo is the last one she needs. When she gets to Cerulean Cave however, she finds out
Elaine had already caught it. She tries to battle Elaine for Mewtwo but loses. After this, she finds out she, Elaine, and
Trace had all been tasked to complete the Pokédex at the same time, so there were no issues at all.