  In \cite{JKS} we gave an (additive) categorification of Grassmannian cluster
algebras, using the category $\CM(A)$ of Cohen-Macaulay modules for a certain
Gorenstein order $A$. In this paper, using a cluster tilting object in the same
category $\CM(A)$, we construct a compatible pair $(B, L)$, which is the data
needed to define a quantum cluster algebra. We show that when $(B, L)$ is
defined from a cluster tilting object with rank 1 summands, this quantum
cluster algebra is (generically) isomorphic to the corresponding quantum
Grassmannian.
