Green will show up a few times during the main story, but it won’t be until the very end that you have a true all-out battle with her.
After defeating the Elite Four, return to Cerulean City and enter the Cerulean Cave. Inside, you will find the elusive Mewtwo. After capturing him, head outside to meet a trainer who tells you that another girl went inside looking for Mewtwo.
Head back to where you found Mewtwo, and Green will be waiting there to battle you.
After you defeat her, she will reward you with Mewtwonite X and Mewtwonite Y to mega-evolve your newly found Mewtwo.
After this event, Green can be found in Cerulean City for subsequent battles. You won’t get any fancy rewards for beating her, but it’s fun to test your strength.