I clearly do not have the same edition of the book as you do, but I think the quote you are talking about is the one at the beginning of Part II where Montag reads the line about friendship.
To me, this shows Clarisse's influence because she gradually kept giving him drops and drops of friendship. No one particular thing that she said really made him stand up and understand what he did not like about their life. But as she kept on being nice to him and giving him ideas, she eventually reached the point where her influence ran over and he understood what she was telling him about their society.
Clarisse is the inciting factor that causes Montag to begin questioning everything he has done and has believed. She represents what is left of good and innocence in Montag's world. She doesn't accept society's rules and values. She serves as the antithesis to Montag's wife who blindly conforms to society. Clarisse supposedly dies when hit by a car, and her family just disappears.