© 2021 Elsevier LtdIn this paper, three equiatomic quaternary Heusler compounds − FeVHfZ (Z = Al, Si, and Ge) − are investigated for their structural, magnetic, electronic, mechanic, and lattice dynamic properties under pressure effect. These compounds are optimized for under three structural types and three magnetic phases: β is the most stable structure with ferromagnetic phase. The electronic properties reveal that FeVHfAl is a half-metal, and that FeVHfSi and FeVHfGe are spin gapless semiconductors. In addition to electronic band structure, possible hybridization and partial density of states are presented. Furthermore, the mechanical properties are studied, and the three-dimensional direction-dependent mechanical properties are visualized under varying pressure effects. Our results reveal the half-metal and spin gapless semiconductor nature of the ferromagnetic FeVHfZ compounds, making them promising materials for spintronics applications.