  The present paper aims to investigate $(m,\rho)$-quasi-Einstein metrices on
almost co-K\"ahler manifolds $\mathcal{M}$. It is proven that if a
$(\kappa,\mu)$-almost co-K\"ahler manifold with $\kappa<0$ is
$(m,\rho)$-quasi-Einstein manifold, then $\mathcal{M}$ represents a
$N(\kappa)$-almost co-K\"ahler manifold and the manifold is locally isomorphic
to a solvable non-nilpotent Lie group. Next, we study the three dimensional
case and get the above mentioned result along with the manifold $\mathcal{M}^3$
becoming an $\eta$-Einstein manifold. We also show that there does not exist
$(m,\rho)$-quasi-Einstein structure on a compact $(\kappa,\mu)$-almost
co-K\"ahler manifold of dimension greater than three with $\kappa<0$. Further,
we prove that an almost co-K\"ahler manifold satisfying $\eta$-Einstein
condition with constant coefficients reduces to a $K$-almost co-K\"ahler
manifold, provided $ma_{1} \neq (2n-1)b_{1}$ and $m \neq 1$. We also
characterize perfect fluid spacetime whose Lorentzian metric is equipped with
$(m, \rho)$-quasi Einstein solitons and acquired that the perfect fluid
spacetime has vanishing vorticity, or it represents dark energy era under
certain restriction on the potential function. Finally, we construct an example
of an almost co-K\"ahler manifold with $(m,\rho)$-quasi-Einstein solitons.
