Quality Factor and Cut-off Frequency Defintion
For second-order circuits (lowpass, bandpass, highpass) the quality factor Q appears in the transfer function as shown in the given bandpass function. 

What is the meaning of the Q-factor?

Answer: It is a measure for the magnitude of the function at the pole frequency wo. (Simply introduce w=wo in the transfer function to see the effect, wn² must read: wo²).
More than that. it is one of two figures which characterize the position of the pole in the left half of the complex s-plane.

The pole frequency wo is nothing else than the length of the pointer from the origin to the actual pole position and the Q-factor (also called "pole-Q") is a measure of the distance to the Im-axis (which is important for the stability margin of the system). 

The factor Q is defined as Q=wo/2*R(p) (R(p)=real part of the pole). From this definition follows that we also have Q=1/2d (with d=damping factor).

(1) Bandpass: It can be shown that this definition for Q gives a value which is equal to the classical bandpass Q=fo/BW (BW: 3dB-bandwidth).

(2) Lowpass and Highpass: There are different Q-factors for the various forms resp. alternatives of the filter (approxinations).

Examples: Q=0.5773 (Thomson-Bessel), Q=0.7071 (Butterworth), Q=0.9565 (Chebyshev, ripple 1 dB), Q=1.3065 (Chebyshev, ripple 3 dB) .