We present a digital quantum computation of two-hadron scattering in a Z2 lattice gauge theory in 1+1 dimensions. We prepare well-separated single-particle wave packets with desired momentum-space wavefunctions, and simulate their collision through digitized time evolution. Multiple hadronic wave packets can be produced using the systematically improvable algorithm of this work, achieving high fidelity with the target initial state, and demanding only polynomial resources in system size. Specifically, employing a trapped-ion quantum computer (IonQ Forte), we prepare up to three meson wave packets\ using 11 and 27 system qubits, and simulate collision dynamics of two meson wave packets for the smaller system. Despite noise effects, post-processing with a global-symmetry-based noise mitigation yields results consistent with numerical simulations, but decoherence limits evolution into long times. We demonstrate the critical role of high-fidelity initial states for precision measurements of state-sensitive observables, such as S-matrix elements and decay amplitudes. While we have not established quantum advantage by this early hardware demonstration, our algorithms are general, and our findings imply the potential of quantum computers in simulating scattering processes in strongly interacting gauge theories.