The definition of Ricci curvature on graphs was given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Math., 2009. Recently, a powerful limit-free formulation of Lin-Lu-Yau curvature using the graph Laplacian has been given in Münch-Wojciechowski, Adv. Math., 2019. Let $F_k$ be the friendship graph obtained from $k$ triangles by sharing a common vertex and $T$ be the graph obtained from a triangle and $K_{1,3}$ by adding a matching between every leaf of $K_{1,3}$ and a vertex of the triangle. In this paper, we classify all the simple connected $C_4$-free graphs with positive Lin-Lu-Yau curvature for minimum degree at least 2: the cycles $C_3,C_5$, the friendship graphs $F_2,F_3$, the line graph of Peterson graph, and $T$.