  Many authors have studied Ricci solitons and their analogs within the
framework of (almost) contact geometry. In this article, we thoroughly study
the $(m,\rho)$-quasi-Einstein structure on a contact metric manifold. First, we
prove that if a $K$-contact or Sasakian manifold $M^{2n+1}$ admits a closed
$(m,\rho)$-quasi-Einstein structure, then it is an Einstein manifold of
constant scalar curvature $2n(2n+1)$, and for the particular case -- a
non-Sasakian $(k,\mu)$-contact structure -- it is locally isometric to the
product of a Euclidean space $\RR^{n+1}$ and a sphere $S^n$ of constant
curvature $4$. Next, we prove that if a compact contact or $H$-contact metric
manifold admits an $(m,\rho)$-quasi-Einstein structure, whose potential vector
field $V$ is collinear to the Reeb vector field, then it is a $K$-contact
$\eta$-Einstein manifold.
