The half-Heusler TiRhSb and ZrRhSb alloys in the formation of face-centered cubic MgAgAs-type structure, which conforms to the F (4) over bar 3m space group with 216 as the space number, have been investigated using Generalized Gradient Approximation (GGA) implemented in Density Functional Theory (DFT). The calculated formation enthalpies and the plotted energy-volume curves of different types of structural phases (alpha, beta, and gamma) in these alloys indicate that the gamma-phase structure is the best energetically suitable structure. In addition, TiRhSb and ZrRhSb alloys have been found as paramagnetic (PM) with the investigation of antiferromagnetic (AFM), ferromagnetic (FM), and paramagnetic (PM) orders in the most stable gamma-phase structure. Therefore, their electronic, mechanical, and dynamical properties have been examined in the gamma structural phase and paramagnetic order. These alloys have semiconducting nature due to the observed same band gaps in both the majority and minority spin channels of the calculated spin-polarised electronic band structure. These calculated band gaps are 0.75 eV for gamma-TiRhSb and 1.18 eV for gamma-ZrRhSb. The predicted elastic constants indicate that the alloys in this study are mechanically stable and show nearly isotropic behavior in the gamma structural phase. Also, the minimum and the diffuson thermal conductivites have been determined for these alloys. Finally, the calculated phonon dispersion spectras for the gamma-TiRhSb and gamma-ZrRhSb half-Heusler antimonide alloys show the dynamic stability of these systems.