Simulations for the 125-GeV Higgs Factory (HF) Muon Collider (MC) have shown large background particle loads on the collider detector. To verify level, source and composition of background calculations were performed using FLUKA and MARS codes for two shielding configurations. After comprehensive tuning of muon beam parameters, geometry setups and scoring procedures, background particle distributions at the detector entrance were simulated and compared. The spatial distributions and energy spectra of background particles obtained by two codes are rather similar. Average numbers of background particles simulated using MARS and FLUKA agree within a factor of two.