  We introduce three non-compact moduli stacks parametrizing noncommutative
deformations of Hirzebruch surfaces; the first is the moduli stack of locally
free sheaf bimodules of rank 2, which appears in the definition of
noncommutative $\mathbb{P}^1$-bundle in the sense of Van den Bergh
arXiv:math/0102005, the second is the moduli stack of relations of a quiver in
the sense of arXiv:1411.7770, and the third is the moduli stack of quadruples
consisting of an elliptic curve and three line bundles on it. The main result
of this paper shows that they are naturally birational to each other. We also
give an Orlov-type semiorthogonal decomposition for noncommutative
$\mathbb{P}^1$-bundles, an explicit classification of locally free sheaf
bimodules of rank 2, and a noncommutative generalization of the (special) McKay
correspondence as a derived equivalence for the cyclic group $\left\langle
\frac{1}{d}(1,1) \right\rangle$.
