Bill Belichick -- All Greece'd Up ... During Tom Brady's Appeal (Pics)

While his superstar QB was fighting the NFL to drop the Deflategate suspension, New England Patriots coach Bill Belichick was chillin' in Greece ... looking like he didn't have a care in the world.

TMZ Sports obtained pics of Bill hangin' out with his GF, Linda Holliday, on the Greek isle of Santorini this week ... doing some shopping, eating and hanging out at a local bar.

We're told the couple spent a lot of time in Oia -- known for being incredibly beautiful ... and one of the most relaxing places on the planet.