Does Q-Factor matter for low pass and high pass filters?
Here's a picture (I drag out now and then) that explains the effect of Q on a 2nd order low pass filter: -



The top three pictures show you the effect of varying the Q-factor. Q-factor can also be reduced to make a maximally flat pass-band (aka a butterworth filter).

The picture goes on to explain where the pole zero diagram comes from and how you can relate natural resonant frequency (\$\omega_n\$) with zeta (\$\zeta\$). For your reference, zeta = 1/2Q.

You will also find that the shape of the curve reverses (with a hump) for 2nd order high pass filters: -



The high-pass filter picture came from here.


  However, low pass and high pass filters do not have centre frequency.


They have the equivalent of a centre frequency known as the natural resonant frequency and if you think about a series L and C making a notch filter: -



This becomes a 2nd order high pass filter if the output is taken from the junction of the capacitor and inductor. Also if L and C swap places, it's still a notch filter but now if you take the output from across C it becomes a 2nd order low pass filter. Same resonant frequency and Q formulas all apply.