The easiest counterexample is $p=2, q=0$. In this case the usual motivic cohomology vanishes, and if $X$ is normal 
$$H^2_{et}(X,\mathbf{Z})=H^1_{et}(X,\mathbf{Q}/\mathbf{Z})= \mathrm{Hom}(\pi_1^{et}(X),\mathbf{Q}/\mathbf{Z})$$
which does not vanish even for fields (which are not separably closed).

Another example is $p=3, q=1$. In this case you get the Brauer group which doesn't vanish in general.