Summary: Let \(C_4\) be a cycle of order 4. Write \(\operatorname{ex}(n,n,n, C_4)\) for the maximum number of edges in a balanced 3-partite graph whose vertex set consists of three parts, each has \(n\) vertices that have no subgraph isomorphic to \(C_4\). In this paper, we show that \(\operatorname{ex}(n,n,n, C_4) \geq \frac{3}{2} n(p + 1)\), where \(n = \frac{p(p - 1)}{2}\) and \(p\) is a prime number. Note that \(\operatorname{ex}(n, n, n, C_4) \leq (\frac{3\sqrt{2}}{2}+ o (1)) n^{\frac{3}{2}}\) from M. Tait and C. Timmons works [J. Comb. Des. 27, No. 6, 391–405 (2019; Zbl 1479.05171)]. Since for every integer \(m\), one can find a prime \(p\) such that \(m \leq p \leq (1+ o(1))m\), we obtain that \(\lim\limits_{n \to \infty} \frac{ex(n,n,n, C_4)}{\frac{3\sqrt{2}}{2} n^{\frac{3}{2}}} = 1\).