The sentence is “correct” as given, but there are some variations that might be stronger. First, [need, I am… ] could be [need; I am..] because the implied “There is” makes it two coordinate clauses [There is no need;] and [I am quite all right] that can be best separated by a semicolon. Also there is some question about the necessity for a comma before the quotation. Finally, the punctuation at the end of the quotation could be an exclamation mark [!] and it would be inside the quotation as part of the utterance. So a somewhat smoother solution would be:
The wanderer found no reason to be wary of him, but nonetheless the proud old man replied “No need; I am quite all right!” as he plodded forward through the street.