In the original Cowboy Bebop series, we meet Faye on Mars during the third episode as a group of thugs is tracking her down. She fires on them with a machine gun but gets captured and brought to their boss named Gordon. He's a casino owner that offers to pay off her debts if she's able to pose as a dealer to get Spike's final chip in a round of blackjack, which is how she meets him and Jet.
In the Netflix version, however, Spike and Jet run into Faye in the first episode while they're on a mission to find a man named Asimov Solensan on the asteroid of Tijuana. She's actually the one who shoots and kills him in this version instead of his wife, Katerina.