We determine the Ramsey numbers \(R(S_{2,m} K_{2, q})\) for \(m \in \{3, 4, 5\}\) and \(q \geq 2\). Additionally, we obtain \(R(tS_{2, 3}, sK_{2, 2})\) and \(R(S_{2, 3}, sK_{2, 2})\) for \(s \geq 2\) and \(t \geq 1\). Furthermore, we also establish \(R(sK_2, \mathcal{H})\), where \( \mathcal{H}\) is the union of graphs with each component isomorphic to the connected spanning subgraph of \(K_{s} + C_n\), for \(n \geq 3\) and \(s \geq 1\).