I might be missing something (seeing that I don't know what a Chow motive is), but I think the answer is yes. It is a result of G.Laumon proved in Comparaison de caractéristiques d’Euler-Poincaré en cohomologie $\ell$-adique, C. R. Acad. Sc. Paris, t. 292 (1981), Série I, 209-212 that for $X$ a separated scheme of finite type over $k$ a field of characteristic $p>0$, the Euler-Poincaré characteristic (over $\bar{k}$) is independent of $\ell≠p$ (the proof consists in showing that the Euler-Poincaré characteristic coincides with the Euler-Poincaré characteristic in compact support; the latter being the degree of the Zeta function and thus independent of $\ell$). Hope this helps.