Consider the projective quadric $V$ given by $$ 2x^2+y^2+z^2+w^2=0 $$ over $\mathbb{Q}$. I will prove that $V$ is not birational to a product of Severi-Brauer varieties over $\mathbb{Q}$.
If $V$ is to be birational to a product $\prod_i W_i$ of Severi-Brauer varieties, then we must have $\dim W_i = 1$ for all $i$ since the $W_i$ have points over a quadratic extension but not all of them have points over $\mathbb{Q}$. So $V \sim W_1 \times W_2$. Also, $V$ has points over $\mathbb{Q}_p$ for $2 < p < \infty$ by Chevalley-Warning and Hensel's lemma, but not over $\mathbb{Q}_2$ and $\mathbb{R}$. So both $W_i$ have points over $\mathbb{Q}_p$ for $2 < p < \infty$, which implies that the $W_i$ are isomorphic to either $\mathbb{P}^1$ or to $W':x^2+y^2+z^2=0$, with not both of them isomorphic to $\mathbb{P}^1$. This then implies that $W_1 \times W_2$ has no points over $\mathbb{Q}(\sqrt{-7})$ (which has a completion isomorphic to $\mathbb{Q}_2$), since the same holds for $W'$, while $V$ contains the point $(1,1,2,\sqrt{-7})$.