  Let $E$ be a vector bundle of rank $n$ on $\mathbb{P}^1$. Fix a positive
integer $d$. Let $\mathcal{Q}(E,d)$ denote the Quot scheme of torsion quotients
of $E$ of degree $d$ and let $Gr(E,d)$ denote the Grassmann bundle that
parametrizes the $d$-dimensional quotients of the fibers of $E$. We compute
Seshadri constants of ample line bundles on $\mathcal{Q}(E,d)$ and $Gr(E,d)$.
