Higher-order topological photonic crystals (TPCs) have attracted much attention in the past several years since they do not obey the conventional bulk-boundary correspondence in topological insulating phases. This characteristic offers a new dimension with which to trap and manipulate the flow of light. In this work, 2D higher-order TPCs for engineering optical second harmonic generation (SHG) are designed. In the TPCs, a topological corner-state is achieved and it is demonstrated that the high localization of photons at the corner-state can significantly promote optical SHG. In addition, topological edge-states are obtained by breaking the Dirac degeneracy point near the SHG frequency. They are also topologically protected and immune to bending resistance. Therefore, the enhanced SHG signal from the corner-state can be manipulated precisely for transmitting through the designed interface with relatively low loss due to the topological-protection edge waves. This design strategy combines the advantages of corner-state and edge-state in higher-order TPCs, demonstrating potential applications of topological photonic devices, such as optical communication and optical computing technologies.