A quantum critical point between the extended and critical phases of the Aubry-Andr\'{e}-Harper model with p-wave pairing, a little exploited because most investigations focus on localization transitions We investigated the scaling properties of the neighborhood. From the critical period to the local period. We find that the spectral average entanglement entropy and the generalized fidelity susceptibility act as prominent universal order parameters for the corresponding critical points without closing the gap. Introducing his Widom scaling ansatz of these critical probes, we develop a unified theory of critical exponents and scaling functions. Therefore, if the Fibonacci sequence increases the size of the system, we extract the critical exponent $\nu$ and the dynamic exponent $z$ of the correlation length through finite-sized scaling. The obtained values of $\nu \simeq 1.000$ and $z \simeq 3.610$ indicate that the expansion-to-critical phase transition belongs to a different universality class than the localization transition. Our approach sets the stage for exploring relevant universal information in unconventional quantum critical and quasi-periodic systems in state-of-the-art quantum simulation experiments.
