Let P be the statement: Every subset of plane belongs to the sigma algebra generated by $\{A \times B : A, B \subseteq \mathbb{R}\}$.

Let Q be the statement: Every sigma algebra on $\mathbb{R}$ of size at most continuum is generated by a countable family.

Both statements are independent of ZFC and P implies Q. Does Q imply P?

This stems from the following question.