In this paper, we introduce $k$-Ricci curvature and $k$-scalar curvature on lightlike hypersurface of a Lorentzian product manifold with quarter-symmetric nonmetric connection. Using these curvatures, we establish some Chen-type inequalities for lighlike hypersurface of a Lorentzian product manifold with quarter-symmetric nonmetric connection. Considering the equality case, we obtain some results.