Griffin picked up the injury when hitting his hand on the helmet of a teammate as he threw a pass. He then went to the locker room for an X-ray before returning to the sideline. Griffin will back up starter Lamar Jackson in 2019.
He signed a two-year extension with the AFC North franchise this off-season.
Griffin, 29, completed 2-of-6 pass attempts in 2018. He had 21 yards through the air in three appearances last season.
"I hit it on the helmet of one of the defenders coming to the line," Griffin said. "I guess that's why they [coaches] tell them to stay away from us [quarterbacks]. We'll figure out what's going on, and I'm sure the team will let you guys know what's up."