Yes. There are multiple isomorphisms between de Rham and $\ell$-adic cohomology. You can get one by combining the de Rham - Betti and Betti - $\ell$-adic isomorphisms. But this is silly as the periods you will get, in $\mathbb Q\otimes_{\mathbb Q} \mathbb Q_\ell$$\mathbb C\otimes_{\mathbb Q} \mathbb Q_\ell$, will just be the usual Betti periods.
A better approach is to use $\ell$-adic Hodge theory, which gives an isomorphism between de Rham cohomology and Betti cohomology when both are tensored to the field $B_{dR}$.
However, in this case, there is no need to use the motivic Tannakian group to study the periods. This is because of the identity $$H^*_{dR} (X, \mathbb Q) \otimes_{\mathbb Q} \mathbb Q_p = (H^*(X, \mathbb Q_\ell) \otimes B_{dR})^{ \operatorname{Gal} (\overline{\mathbb Q_\ell} | \mathbb Q_\ell )} $$$$H^*_{dR} (X, \mathbb Q) \otimes_{\mathbb Q} \mathbb Q_\ell = (H^*(X, \mathbb Q_\ell) \otimes B_{dR})^{ \operatorname{Gal} (\overline{\mathbb Q_\ell} | \mathbb Q_\ell )} $$ which implies that the periods in $B_{dR}$ are determined by the $\ell$-adic Galois representation restricted to $\operatorname{Gal} (\overline{\mathbb Q_\ell} | \mathbb Q_\ell )$ and thus are related to the Tannakian group of the category of $\operatorname{Gal} (\overline{\mathbb Q_\ell} | \mathbb Q_\ell )$-representations. This Tannakian group can be much smaller than the motivic Galois group.