Summary: Let \(G\) be a graph and \(H\) a spanning subgraph of \(G\). A backbone-\(k\)-coloring of \((G, H)\) is a mapping \(f: V(G)\to \{1,2,\ldots, k\}\) such that \(|f(u)-f(v)|\geqslant 2\) if \(uv\in E(G)\) and \(|f(u)-f(v)|\geqslant 1\) if \(uv\in E(G)\backslash E(H)\). The backbone chromatic number of \((G, H)\), denoted by \(\chi_b(G, H)\), is the smallest integer \(k\) such that \((G, H)\) has a backbone-\(k\)-coloring. In this paper, we prove that if \(G\) is a connected \(C_4\)-free planar graph, then there exists a spanning tree \(T\) of \(G\) such that \(\chi_b(G, T)\leqslant 4\).