To advance the understanding of plasma material interactions, the Material Plasma Exposure eXperiment (MPEX) is a new linear plasma device that will generate and deliver plasma relevant to future fusion reactor divertors. The operation of MPEX is planned to be steady-state in order to facilitate high fluence exposures of plasma facing materials and components. The desire for steady-state operation along with the magnetic field requires the utilization of superconducting coils. The superconducting magnet system for MPEX has been developed. The baseline model has six superconducting magnet and one room-temperature magnet subsystems. In order to protect multiple superconducting magnet systems, quench analysis was carried out to determine the best protection approach for each magnet type. Because the mutual inductance accounts for approximately 35% of the stored energy in the entire system, this must be considered when determining the peak voltages and temperatures during a quench. Two approaches for passive quench protection are considered: (1) self-protecting magnets and (2) use of diodes to sub-divide the coils. For both approaches, active quench detection will be used to ensure all coils are de-energized in the event of a quench. Results of the quench analysis for several quench scenarios are presented.