The structural, electronic, magnetic, and thermoelectric properties of CoFeCuZ (Z = Al, As, Ga, In, Pb, Sb, Si, Sn) quaternary Heusler alloys were investigated using density functional theory. The calculated formation energies confirmed that these alloys are thermodynamically stable. The CoFeCuPb alloy is predicted to be a half-metallic ferromagnet under the equilibrium lattice constant with a spin-down band gap and a total magnetic moment of 0.303 eV and 4.0μB, respectively. However, the other alloys are either metallic for Z = Al, As or nearly half-metallic for Z = Ga, In, Sb, Si, Sn. It was found that CoFeCuPb is half-metallic under uniform pressure that ranges from −12.75 GPa to 8.46 GPa with an optimum band gap at the equilibrium lattice constant. The total magnetic moment of CoFeCuPb was robust under pressure that ranges between −6.3 GPa and 13.89 GPa. The thermoelectric properties are also investigated for CoFeCuPb alloy using classical transport theory. Under an equilibrium lattice constant, high power factors of 159.5 × 1014 μWcm−1 K−2 s−1 and 69.5 × 1014 μWcm−1 K−2 s−1 are obtained at 800 K and 300 K, respectively.