As I think Nacho has mentioned, "¡Qué [... x ...] ni qué ocho cuartos!" is an expression to say that the thing (or idea) that is mentioned just preceeding it (where the "x" is) is "nonsense".
So perhaps in English we would translate the sentence you gave as follows:
"Did you say 'permission'? ¡Nonsense! Not 'permission' nor any other (similar) thing, I say! It's my permission that they will never get!"
As for the exact meaning of "ocho cuartos", this in fact no longer exists in Spain---but let me explain! :) Before using "euro" as our currency, we had "pesetas" in Spain; and one "peseta" (one coin of that currency) was divided into 4 "cuartos" (quarters). So, as you can guess, "ocho cuartos" equals two pesetas.
Apparently, in that time when "ocho cuartos" were used to pay for goods (e. g. a pair of shoes), one day someone must have said they considered the price of something (e. g., that pair of shoes) too high... ridiculous... nonsense. And so that person must have said in disagreement:
"What? You want 3 pesetas for that pair of shoes! Not 3 pesetas nor ocho cuartos [i. e. 2 pesetas] will I give you, I say!"
So, from that moment on, the expression "¡Qué [... x ...] ni qué ocho cuartos!" must have become popular to show that you disagree or consider something (previously said) "nonsense".
Interesting story!---though I never use the expression myself, anyway (I'm not that old)! :)
I hope that helps!