Properties on conditional demand correspondence from the textbook of Mas-Colell et al
Let $z_1$ and $z_2$ be $\geq 0$ and solution to
$$\min_z \{w^\top z\lvert f(z)\geq q\}$$
then clearly $f(z_1)\geq q$ and $f(z_2)\geq q$ and since $\{z\geq 0\lvert f(z)\geq z \}$ is convex it then follows that $z_3 := \lambda z_1 + (1-\lambda)z_2$ must satisfy the constraint $f(z_3)\geq q$.
Since $z_1$ and $z_2$ are both minimizers it cannot be the case that $w^\top z_1 \not = w^\top z_2$ rather it must be the case that $w^\top z_1 = w^\top z_2$. This implies that $w^\top z_3 = w^\top(\lambda z_1 + (1-\lambda)z_2) = \lambda w^\top z_1 + (1-\lambda) w^\top z_2 = w^\top z_1$ hence $z_3$ is a minimizer.
Since $z_3$ is minimizer and satisfy constraint it must be in $z(w,q)$ which therefore must be convex set.
If $f$ is quasi concave then by definition $\{z\geq 0 \lvert f(z) \geq q\}$ is convex and therefore using the above argument implies $z(w,q)$ is convex.