The easiest counterexample is $p=2, q=0$. In this case the usual motivic cohomology vanishes, and if $X$ is normal $$H^2_{et}(X,Z)=H^1_{et}(X,Q/Z)= Hom(\pi_1(X),Q/Z)$$ which does not vanish even for fields (which are not separably closed).
Another example is and p=3, q=1. In this case you get the Brauer group which doesn't vanish in general.