The constraint qualification Q-CCCQ plays an important role in quasiconvex programming and has been developed by many authors to investigate the set containment problem, duality and optimality conditions for quasiconvex programming. In this paper, we consider an infinite quasiconvex inequality system defined by a family of proper lower semicontinuous quasiconvex functions $\{h_i : i \in I \}$ and establish some sufficient conditions for ensuring the Q-CCCQ in terms of the interior-point condition together with approximate continuity assumption of the function $i \mapsto h_i(x)$.