In this work, we use the Green's function technique and investigate transport properties of the electronic states in the two-dimensional (2D) second-order topological insulator (TI). We calculated the bulk, edge, and corner states corresponding respectively to the infinite plane, one-dimensional nanoribbon, and the square flake geometries of the 2D second-order TI. Existence of the corner states in the square flake is confirmed. In this approach, we considered tunneling via the localized corner states in the double-barrier-modulated 2D second-order TI. Sharp peaks are found in the transmission probability and conductance incurred by the corner states lying within the surrounding energy gap. The relation between transmission peaks and the corner states is confirmed by their energy correspondence. The extremely discrete and sharp resonance might lend insight to experimental observation of the corner states in the 2D second-order TI.