Young and Thigpen both participated in the Browns' three-day minicamp, which started Tuesday.
Young was not in the league last season, and Farmer said earlier this week that he wanted to see Young perform and that the minicamp tryout was the best way.
The Browns needed to add a veteran to back up Brian Hoyer, who is penciled in as the starter heading into this season but is coming off knee surgery. Other than Hoyer, the only quarterback on Cleveland's roster was Alex Tanney, who was signed late last season as the Browns were hit hard by injuries at the position.
Young last played in the NFL for the Eagles in 2011, though he was with the Packers during the preseason in 2013. Thigpen played in four games for Buffalo in 2012.
Both fit the play-action rollout system that offensive coordinator Kyle Shanahan has run.
Thigpen has played in 29 games in six seasons, completing 54 percent of his passes with 21 touchdowns and 18 interceptions.
Young made 50 starts from 2006 to 2011. He completed 57.9 percent of his passes with 46 touchdowns and 51 interceptions. He is 31-19 as a starter and twice made the Pro Bowl.
The Browns on Thursday also signed linebacker Zac Diles, who spent parts of the past two seasons with Tennessee.
Information from ESPN.com Browns reporter Pat McManamon and The Associated Press was used in this report.