Another fictional hawk could be 'Hawkeye' which is a name that can be associated with a hawk - like character. It gives the impression of a character with sharp vision and a focused, predatory nature, much like a real hawk.
The Hawks' back story could involve their origin as a sports team. They might have started in a particular city with a group of passionate players and investors. Maybe they were initially part of a smaller league and then grew to be in a major one. Their name could be inspired by the characteristics associated with hawks, like speed and precision.