Is there a quote in "To Kill a Mockingbird" that proves it is a bildungsroman?
Tricky question! IS there a single accepted "quote" that 'proves' "coming of age"? I did not recall any, nor find one in review.
The simple English for "bildungsroman" is "coming of age story". That is a process in any case, and may be the main theme or a side theme of a novel.
Rarely does a good story have such an explicit statement as "and then she became an adult".
"To Kill a Mockingbird" IS considered a "bildungsroman", but not on the basis of a single quote.
"To Kill a Mockingbird can be read as a feminist Bildungsroman, for Scout emerges from her childhood experiences with a clear sense of her place in her community and an awareness of her potential power as the woman she will one day be."
Ref: Ware, Michele "'Just a Lady': Gender and Power in Harper Lee’s To Kill a Mockingbird" in Women in Literature: Reading Through the Lens of Gender Jerilyn Fisher and Ellen S. Silber (eds.), Greenwood Press (2003). ISBN 978-0-313-31346-2.