  We have investigated scaling properties near the quantum critical point
between the extended phase and the critical phase in the Aubry-Andr\'{e}-Harper
model with p-wave pairing, which have rarely been exploited as most
investigations focus on the localization transition from the critical phase to
the localized phase. We find that the spectrum averaged entanglement entropy
and the generalized fidelity susceptibility act as eminent universal order
parameters of the corresponding critical point without gap closing. We
introduce a Widom scaling ansatz for these criticality probes to develop a
unified theory of critical exponents and scaling functions. We thus extract the
correlation-length critical exponent $\nu$ and the dynamical exponent $z$
through the finite-size scaling given the system sizes increase in the
Fibonacci sequence. The retrieved values of $\nu \simeq 1.000$ and $z \simeq
3.610$ indicate that the transition from the extended phase to the critical
phase belongs to a different universality class from the localization
transition. Our approach sets the stage for exploring the unconventional
quantum criticality and the associated universal information of quasiperiodic
systems in state-of-the-art quantum simulation experiments.
