Is this sentence:
What kind of a mother would say, “That backpack looks ugly on you.”?
I read the AP StyleBook’s section on this and, yes–it is correctly punctuated.
From APStylebook, accessed 10-3-2019, 7:37 am, It says,
“Using parentheses with other punctuation marks: 1. When a complete sentence within parentheses stands alone (that is, not as part of another sentence), its end punctuation goes inside the closing parenthesis.”
While the question mark, that punctuates the whole sentence goes outside of the inner sentence’s, divided by quotation marks.