 Solution

Let $D\in M$ be a dense subset of a forcing order $\mathbb{P}$. Then $D\times \mathbb{Q}\in M$ is dense in the product forcing order: below $(p,q)$, choose $p^{\prime }\le p$ in $D$ and retain $q$. The generic filter $H$ meets it, so its first projection $G_0$ meets $D$. Together with the filter properties proved above, this givesThe same argument proves that $G_1$ is $\mathbb{Q}$-generic over $M$, but genericity over the larger generic extension $M[G_0]$ needs the next argument.