Let $h:qCat \rightarrow Cat$ be the fundamental/homotopy category functor, which is the left adjoint functor to the nerve functor $N$ which sends a category to simplicial $N C_n=Hom([n],C)$.
I believe that $h$ commtues with products. Is that true? I used the explicit representation of homotopy category of quasicategories.
Yes, this works for arbitrary products. The point is that $hQ$ is simply a quotient of the graph $Q_1\rightrightarrows Q_0$ under the homotopy relation; the mapping of $Q$ to its graph of arrows and objects of course preserves all products, and so does the homotopy relation, since a homotopy $(f_i)\sim (g_i)$ between edges in some product $\prod Q_i$ is by definition a 2-simplex in the products whose projections are homotopies $f_i\sim g_i$.
You may know that $h$ is the restriction of the much less manageable left adjoint $\tau_1: \text{Sset}\to \text{Cat}$. It is nontrivial to prove that $\tau_1$ preserves even finite products, though this has been well known since Gabriel and Ulmer. This result is very important, for instance in Thomason's theorem establishing the equivalence of model categories of simplicial sets and small categories, but does not generalize to infinite products. $\tau_1$ is a quotient of the free category on the underlying graph of a simplicial set, and this step of taking a free category ruins commutation with infinite products. An explicit counterexample comes, if I recall correctly, from taking the product of the spines of the $n$-simplices over all $n$.