After throwing an interception in Monday night's 16-0 loss to the Ravens, Quinn dived at Suggs' knees while trying to bring down cornerback Chris Carr, who had picked him off. Suggs had to leave the game and could miss significant playing time.
Ravens linebacker Ray Lewis accused Quinn of a cheap shot.
Quinn understands why the Ravens were mad and can appreciate them questioning his intentions.
"I can see why they'd be upset," he said. "But again, he wasn't even in my vision. I was trying to get to the ball. He cut across my face as I was already trying to jump down for the tackle."
Quinn dropped his head and threw his body into Suggs' right knee, thinking he would be able to bring down Carr. He was flagged for a low block and the Browns were penalized 15 yards.
Quinn received a letter from the league on Wednesday but would not disclose the size of his fine, saying only that it was "a good amount."
He maintains he was not trying to hurt Suggs, with whom he has worked out in the offseason. Quinn apologized to him and the Ravens after the game and plans to reach out to Suggs again.
"I wasn't trying to go for him," he said. "I was trying to go for the ball carrier. Unfortunately, a thing like that can happen. I'm praying for him. Hopefully he'll be all right."
Browns coach Eric Mangini believes Quinn was only trying to tackle Carr.
"Knowing Brady, he's a good person," Mangini said. "I don't think he would ever look to hurt someone deliberately."