Alex Smith's Wife Praises QB's Leg Recovery On Birthday, 'Refuses To Give Up'

Alex Smith's wife is celebrating the QB's 36th birthday by showing just how hard he's working to get back on the field ... and praising him for refusing to give up after his life-threatening leg injury.

It's been a roller coaster 18 months for the Smiths after the Redskins QB broke his leg during a game in November 2018 ... an injury that required 17 follow-up surgeries due to a flesh-eating bacteria.

Despite being told it would be nearly impossible to return to football, Smith has been working out like crazy to prove everyone wrong ... and Elizabeth gave an inside look at his progress on Thursday,

"Happy Happy Birthday to this handsome, hard working man❤️" Elizabeth said on IG ... while thanking fans for their support following the debut of ESPN's "Project 11" documentary.

"The last week we’ve been overwhelmed with love, support and sweet messages. We cannot thank you enough."

Elizabeth shared several shots of Smith crushing workouts on his repaired leg (with a bit of help from his 2 sons) ... from cone drills to throwing -- and the dude looks GOOD.

"#project11 was a look into what the last 18 months have been (mostly Alex refusing to give up)." Elizabeth continued.

"I am in awe of his perseverance everyday. He just keeps getting stronger and refuses to give up... Forever grateful"

As for Alex, he knows his NFL future is uncertain ... but he's made it clear he's not giving up without a fight.