In the Stern-Volmer equation the intensity ratio is I0/I = 1+ kqτ0[Q]. This means that also the second term on the rhs has to be dimensionless. Expressing [Q] in different units implies you have to express the quenching rate kq in different units as well. Of course it is an almost trivial exercise to convert molar concentration to ppm or mg/ml or the other way around. I would suggest you do that.
In the Stern-Volmer equation the intensity ratio is I0/I = 1+ kqτ0[Q]. This means that also the second term on the rhs has to be dimensionless. Expressing [Q] in different units implies you have to express the quenching rate kq in different units as well. Of course it is an almost trivial exercise to convert molar concentration to ppm or mg/ml or the other way around. I would suggest you do that.
It's not mandatory that [Q] to be in moles but as Prof. Gert Van der Zwan said as Q unit changes accordingly kq unit also changes. And also for comparison of kq with known values, it's customary to express in moles.