Let $$ex(n, C_4)$$ e x ( n , C 4 ) denote the maximum size of a $$C_4$$ C 4 -free graph of order $$n$$ n . For an even integer or odd prime power $$q$$ q , we prove that $$ex(q^2+q+2,C_4)<\frac{1}{2}(q+1)(q^2+q+2)$$ e x ( q 2 + q + 2 , C 4 ) < 1 2 ( q + 1 ) ( q 2 + q + 2 ) , which leads to an improvement of the upper bound on Ramsey numbers $$R(C_4,W_{q^2+2})$$ R ( C 4 , W q 2 + 2 ) , where $$W_n$$ W n is a wheel of order $$n$$ n . By using a simple polarity graph $$G_q$$ G q for a prime power $$q$$ q , we construct the graphs whose complements do not contain $$K_{1,m}$$ K 1 , m or $$W_m$$ W m , and then determine some exact values of $$R(C_4,K_{1,m})$$ R ( C 4 , K 1 , m ) and $$R(C_4,W_{m})$$ R ( C 4 , W m ) . In particular, we prove that $$R(C_4,K_{1, q^2-2})=q^2+q-1$$ R ( C 4 , K 1 , q 2 - 2 ) = q 2 + q - 1 for $$q\ge 3$$ q ≥ 3 , $$R(C_4,W_{q^2-1})=q^2+q-1$$ R ( C 4 , W q 2 - 1 ) = q 2 + q - 1 for $$q\ge 5$$ q ≥ 5 , and $$R(C_4,W_{q^2+2})=q^2+q+2$$ R ( C 4 , W q 2 + 2 ) = q 2 + q + 2 for $$q\ge 7$$ q ≥ 7 .