Finding cliques in a graph has several applications for its pattern matching ability. $k$-clique problem, a special case of clique problem, determines whether an arbitrary graph contains a clique of size $k$, has already been addressed in quantum domain. A variant of $k$-clique problem that lists all cliques of size $k$, has also popular modern-day applications. Albeit, the implementation of such variant of $k$-clique problem in quantum setting still remains untouched. In this paper, apart from theoretical solution of such $k$-clique problem, practical quantum gate-based implementation has been addressed using Grover's algorithm. This approach is further extended to design circuit for the maximum clique problem in classical-quantum hybrid architecture. The algorithm automatically generates the circuit for any given undirected and unweighted graph and any given $k$, which makes our approach generalized in nature. The proposed approach of solving $k$-clique problem has exhibited a reduction of qubit cost and circuit depth as compared to the state-of-the-art approach, for a small $k$ with respect to a large graph. A framework that can map the automated generated circuit for clique problem to quantum devices is also proposed. An analysis of the experimental results is demonstrated using IBM's Qiskit.