Consider the projective quadric $V$ given by
$$
2x^2+y^2+z^2+w^2=0
$$
over $\mathbb{Q}$. Inspired by Jason Starr's remark on splitting fields, 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 $W':x^2+y^2+z^2=0$, since these are the only conics over $\mathbb{Q}$ up to isomorphism that have points over $\mathbb{Q}_p$ for all $2 < p < \infty$; moreover, not both $W_i$ can be 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 $(1,1,2,\sqrt{-7})$.