Nested Hilbert Schemes on Hirzebruch surfaces and quiver varieties

For $n\ge 1$ we show that the length 1 nested Hilbert scheme of the total space $X_n$ of the line bundle ${\mathcal{O}}_{{\mathbb{P}}^1}(-n)$, parameterizing pairs of nested 0-cycles in $X_n$, is a quiver variety associated with a suitable quiver with relations. This generalizes previous work about nested Hilbert schemes on ${\mathbb{C}}^2$ in one direction, and about the Hilbert schemes of points of $X_n$ in another direction.