"My whole goal is preparing myself and getting ready to try to take over the starting job and lead this team," Quinn said Saturday during an appearance at an auto show.
Anderson agreed to his new deal Friday, a move that took him off the restricted free-agent market. The contract includes $14.5 million guaranteed.
General manager Phil Savage says Anderson is the clear starter entering training camp. Anderson threw 29 TD passes last season and led the Browns to 10 wins.
Quinn understands the situation and even applauded Anderson's signing, but doesn't plan on carrying a clipboard again. His mind-set is "just competing and try to take this team to the playoffs, something we weren't able to do last year."
The Browns traded their first-round pick in 2008 for the chance to draft Quinn No. 22 last April.
He looked good in preseason after holding out, then spent the final 15 weeks of the regular season as Anderson's backup and didn't throw a pass until the finale, when he went 3-for-8 for 45 yards.
Coach Romeo Crennel will open camp with Anderson the starter, but maintained competition breeds success.
"Both of those guys are going to compete," he said Saturday at team headquarters in Berea. "And then we'll see who gives us the best chance."