Summary: 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 \({\mathbb{R}}^{n+1}\) 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.