  Let $C$ be a smooth curve over an algebraically closed field $\mathbf{k}$,
and let $E$ be a locally free sheaf of rank $r$. We compute, for every $d>0$,
the generating function of the motives $[\mathrm{Quot}_C(E,\boldsymbol{n} )]
\in K_0(\mathrm{Var}_{\mathbf{k}})$, varying $\boldsymbol{n} = (0\leq
n_1\leq\cdots\leq n_d)$, where $\mathrm{Quot}_C(E,\boldsymbol{n} )$ is the
nested Quot scheme of points, parametrising $0$-dimensional subsequent
quotients $E \twoheadrightarrow T_d \twoheadrightarrow \cdots
\twoheadrightarrow T_1$ of fixed length $n_i = \chi(T_i)$. The resulting
series, obtained by exploiting the Bialynicki-Birula decomposition, factors
into a product of shifted motivic zeta functions of $C$. In particular, it is a
rational function.
