Bill Belichick was grinning ear-to-ear the day he first met his now-girlfriend Jordon Hudson ... check out a photo TMZ Sports has obtained from their first encounter -- the ex-New England coach was clearly giddy.
The pic was taken at some point during a February 2021 flight -- after the future Hall of Famer had sat next to Hudson as they were both headed from the Boston area to Florida.
Our sources say at one point during the trip ... Belichick leaned in to inquire about some school work Hudson -- a college student at the time -- had been working on, and they hit it off.
We're told the two talked about Hudson's "Deductive Logic" textbook for a while -- and the conversation was so good, Belichick ultimately autographed the inside cover of it.
Alongside his John Hancock, Belichick wrote to the now 23-year-old, "Thanks for giving me a course on logic!"
The two at some point exchanged contact information ... and our sources say they remained friendly following the flight. Eventually, after 72-year-old Belichick broke things off with his longtime girlfriend Linda Holliday, they became romantic.
The two have kept their relationship largely under wraps ... although, recently, they've been seen together in Croatia and at Tom Brady's Patriots Hall of Fame induction ceremony.
Neither has commented publicly on their relationship.