  We exhibit quantum cluster algebra structures on quantum Grassmannians
$K_q[Gr(2,n)]$ and their quantum Schubert cells, as well as on $K_q[Gr(3,6)]$,
$K_q[Gr(3,7)]$ and $K_q[Gr(3,8)]$. These cases are precisely those where the
quantum cluster algebra is of finite type and the structures we describe
quantize those found by Scott for the classical situation.
