We present a general computational framework to investigate ground state properties of quantum spin models on infinite two-dimensional lattices using automatic differentiation-based gradient optimization of infinite projected entangled-pair states. The approach exploits the variational uniform matrix product states to contract infinite tensor networks with unit-cell structure and incorporates automatic differentiation to optimize the local tensors. We applied this framework to the Kitaev-type model, which involves complex interactions and competing ground states. To evaluate the accuracy of this method, we compared the results with exact solutions for the Kitaev model and found that it has a better agreement for various observables compared to previous tensor network calculations based on imaginary-time projection. Additionally, by finding out the ground state with lower variational energy compared to previous studies, we provided convincing evidence for the existence of nematic paramagnetic phases and 18-site configuration in the phase diagram of the K-Γ model. Furthermore, in the case of the realistic K-J-Γ-Γ′ model for the Kitaev material α-RuCl3, we discovered a non-colinear zigzag ground state. Lastly, we also find that the strength of the critical out-of-plane magnetic field that suppresses such a zigzag state has a lower transition field value than the previous finite-cylinder calculations. The framework is versatile and will be useful for a quick scan of phase diagrams for a broad class of quantum spin models.