Don't know about quartertones, but normal harps work like this - they have 7 pedals, one for each C, D, E, F, G, A and B. Each pedal has three positions neutral, 1/2 tone up, and 1/2 tone down. If you can find a way to get your desired scale bu adjusting the 7 pedals, then yes, the harp can glissando it, otherwise it can't. For example. Suppose I want to gliss the chord B-D-E-A-flat. Then I would configure the pedals like this:
C - 1/2 tone down
D - neutral
E - 1/2 tone up
F - neutral
G - 1/2 tone up
A - 1/2 tone down
B - neutral
Hope this helped