The electronic and thermoelectric properties of four ternary chalcogenides with space group Pnma, namely, Cu(Sb,Bi)(S,Se)2, are investigated up to 8 GPa hydrostatic pressure using density functional theory combined with semiclassical Boltzmann theory. The effects of the van der Waals interaction are included in all calculations, since these compounds have layered structures. They all have indirect band gaps that decrease monotonically with increasing hydrostatic pressure except for CuBiS2, for which an indirect-indirect band gap transition occurs around 3 GPa, leading to conduction band convergence with a concomitant 20% increase in the Seebeck coefficient. The enhanced Seebeck coefficient results from a complex interplay between multivalley and multiband effects as well as changes of the band effective masses, driven by hydrostatic pressure. Our results suggest that ongoing developments in high-pressure science may open new opportunities for the discovery of efficient thermoelectric materials.