Using a cyclotron-based model problem, we demonstrate for the first time the applicability and usefulness of an uncertainty quantification (UQ) approach in order to construct surrogate models. The surrogate model quantities, for example, emittance, energy spread, or the halo parameter, can be used to construct a global sensitivity model along with error propagation and error analysis. The model problem is chosen such that it represents a template for general high-intensity particle accelerator modeling tasks. The usefulness and applicability of the presented UQ approach is then demonstrated on an ongoing research project, aiming at the design of a compact high-intensity cyclotron. The proposed UQ approach is based on polynomial chaos expansions and relies on a well-defined number of high-fidelity particle accelerator simulations. Important uncertainty sources are identified using Sobol' indices within the global sensitivity analysis.