The \textit{square} of a graph $G$, denoted by $G^2$, is obtained from $G$ by adding an edge to connect every pair of vertices with a common neighbor in $G$. In this paper we prove that for every planar graph $G$ with maximum degree at most $5$, $G^2$ admits a proper vertex coloring using at most $17$ colors, which improves the upper bound $18$ recently obtained by Hou, Jin, Miao, and Zhao.