In this paper, we study generalized quasi-Einstein manifolds (Mn, g, V, λ) satisfying certain geometric conditions on its potential vector field V whenever it is harmonic, conformal, and parallel. First, we construct some integral formulas and obtain some triviality results. Then, we find some necessary conditions to construct a quasi-Einstein structure on (Mn, g, V, λ). Moreover, we prove that for any generalized Ricci soliton (M̄=I×fM,ḡ,ξ̄,λ), where ḡ is a generalized Robertson–Walker spacetime metric and the potential field ξ̄=h∂t+ξ is conformal, (M̄,ḡ) can be considered as the model of perfect fluids in general relativity. Moreover, the fiber (M, g) also satisfies the quasi-Einstein metric condition. Therefore, the state equation of (M̄=I×fM,ḡ) is presented. We also construct some explicit examples of generalized quasi-Einstein metrics by using a four-dimensional Walker metric.