Beckett, now with the Dodgers, no-hit the Phillies at Citizens Bank Park. Redmond said he sent Beckett a message, praising him for his effort.
"He's not the kind of guy who really looks at the stats and all that stuff," Redmond said. "But with all the big games that he pitched, he definitely deserved a no-hitter. It was cool. I wish I could have been out there to catch him."
The Marlins have faced Beckett twice this season, splitting the two games.
Beckett, now 34, is in the twilight of his impressive career, which includes a World Series MVP Award with the Marlins in 2003, plus a 20-win season and a second World Series championship with the Red Sox in 2007.
Recently, Redmond spoke with Beckett about the pitcher's future.
"We were talking about just baseball and how many years he's played and how he feels," Redmond said. "When you start talking about maybe your career ending and whatever decision he makes, I told him, I said, 'Hey, man, you never know, this might be your best year. This might be the year you have the most fun because you have no expectations on yourself. You go out there, pitch every five days.'"