  Let $X$ be a smooth projective surface over an algebraically closed field $k$
such that $char(k) \neq 2$. Let $X^{[d]}$ denote the punctual Hilbert scheme of
zero dimensional quotients of degree $d$ and $X^{(d)}$ denote the symmetric
product of $X$. For $\ell \neq 2$, we give a formula for the $\ell$-primary
part of the Brauer group of $X^{[2]}$. We show that the Hilbert to Chow
morphism induces an isomorphism of cohomological Brauer groups for $d=2$ and a
similar result for $d \geq 3$. Let $Q(r,d)$ denote the punctual Quot-scheme
parametrising zero dimensional quotients of $\mathcal{O}_X^{ \oplus r}$ of
degree $d$. We show that the natural morphism from $Q(r,d) \rightarrow X^{[d]}$
induces an isomorphism on cohomological Brauer groups.
