  In this paper, we address the integration problem of the isomonodromic system
of quantum differential equations ($qDE$s) associated with the quantum
cohomology of $\mathbb P^1$-bundles on Fano varieties. It is shown that bases
of solutions of the $qDE$ of the total space of the $\mathbb P^1$-bundle can be
reconstructed from the datum of bases of solutions of the corresponding $qDE$
associated with the base space. This represents a quantum analog of the
classical Leray-Hirsch theorem in the context of the isomonodromic approach to
quantum cohomology. The reconstruction procedure of the solutions can be
performed in terms of some integral transforms, introduced in arXiv:2005.08262,
called $Borel$ $(\alpha,\beta)$-$multitransf\!orms$. We emphasize the
emergence, in the explicit integral formulas, of an interesting sequence of
special functions (closely related to iterated partial derivatives of the
B\"ohmer-Tricomi incomplete Gamma function) as integral kernels. Remarkably,
these integral kernels have a universal feature, being independent of the
specifically chosen $\mathbb P^1$-bundle. When applied to projective bundles on
products of projective spaces, our results give Mellin-Barnes integral
representations of solutions of $qDE$s. As an example, we show how to integrate
the $qDE$ of blow-up of $\mathbb P^2$ at one point via Borel multitransforms of
solutions of the $qDE$ of $\mathbb P^1$.
