Technically, the only quote in the book that directly connects Boo Radley to a mockingbird is in Ch. 30.
Sheriff Tate has told Atticus he will not be pressing charges against anyone, Jem or Boo, for stabbing Bob Ewell. It is obvious that Boo was the one who did stab Bob Ewell, although Atticus at first mistakenly thought it was Jem.
When Sheriff Tate leaves, Atticus tells Scout that Bob Ewell simply fell on his knife, and she replies with,
"Yes, sir, I understand...Mr. Tate was right."
Atticus is slightly confused and asks her what she means and she replies with,
"Well, it'd be sort of like shootin' a mockingbird, wouldn't it?"
What she means by this is that Boo Radley is innocent...not of stabbing Bob Ewell, but in general. He is innocent in nature. There would be no benefit to publicly acknowledging Boo was the one to stab Bob Ewell. This would only bring Boo into a spotlight which he has never been in, spawn more rumors, validate other ones, and for what purpose? He would probably not even be charged, as he was only protecting the children. Putting him through that public scrutiny, however, is why Scout said it would be like shooting a mockingbird. It would be uncomfortable and painful, even, for Boo. Sheriff Tate knows this and decides to make up a different story to avoid putting Boo through any of that.
Probably the most memorable quote connecting Boo with the mockingbird comes at the end of the story when Sheriff Tate decides to call Bob Ewell's death self-inflicted--effectively keeping Boo from having to undergo the publicity of an inquest or trial. Scout agrees with Tate's decision, telling Atticus that doing so would
"... be sort of like shootin' a mockingbird, wouldn't it?"