  Let $X$ be a compact connected Riemann surface, and let ${\mathcal Q}(r,d)$
denote the quot scheme parametrizing the torsion quotients of ${\mathcal
O}^{\oplus r}_X$ of degree $d$. Given a projective structure $P$ on $X$, we
show that the cotangent bundle $T^*{\mathcal U}$ of a certain nonempty Zariski
open subset ${\mathcal U}\, \subset\, {\mathcal Q}(r,d)$, equipped with the
natural Liouville symplectic form, admits a canonical deformation quantization.
When $r\,=\,1\,=\, d$, then ${\mathcal Q}(r,d)\,=\, X$; this case was addressed
earlier in \cite{BB}.
