For n greater or equal than 1 we show that the length 1 nested Hilbert scheme of the total space Xn of the line bundle OP1(-n), parameterizing pairs of nested 0-cycles in Xn, is a quiver variety associated with a suitable quiver with relations. This generalizes previous work about nested Hilbert schemes on C2 in one direction, and about the Hilbert schemes of points of Xn in another direction.