A Killing submersion is a Riemannian submersion from a 3-manifold $\mathbb{E}$ to a surface $M$, both connected and orientable, whose fibers are the integral curves of a non-vanishing Killing vector field $\xi\in\mathfrak{X}(\mathbb{E})$. In this setting we give a suitable definition of the graph of a function $f\in C^\infty(U)$, where $U$ is an open subset of $M$. We study the existence and uniqueness of solutions for the Jenkins-Serrin problem on relatively compact domains of $M$ and we prove two general Collin-Krust type estimates for prescribed mean curvature graphs that extend classical result. Finally, we use these tools to prove that in the Heisenberg group there exists a unique minimal graph with prescribed bounded boundary values in every unbounded domain contained in a strip of the plane $\left\{z=0\right\}$. This talk is partially based on a joint work with J.M. Manzano and B. Nelli.