By using the first-principles method, the electronic structures and magnetism of equiatomic quaternary Heusler alloys FeRhCrSi and FePdCrSi were calculated. The results show that both FeRhCrSi and FePdCrSi compounds are ferrimagnets. Both compounds are half-metals and their half-metallicity can be maintained in a wide range of variation of the lattice constant under hydrostatic strain and c/a ratio range under tetragonal distortion, implying that they have low sensitivity to external interference. Furthermore, the total magnetic moments are integers, which are typical characteristics of half-metals. The calculated negative formation energy and cohesive energy indicate that these two alloys have good chemical stability. Furthermore, the value of the elastic constants and the various moduli indicate the mechanical stability of these two alloys. Thus, FeRhCrSi and FePdCrSi are likely to be synthesized in the experiment.