We present an uncertainty quantification framework for operator learning, Deep Operator Gaussian Process (DeepOGP), which generalizes the standard GP in function approximation to operator learning tasks. DeepOGP consists of two independent learning modules. One is designed to learn the features representing uncertainties in function level, which corresponds to the deep kernel learning for GP. The other is to represent uncertainties in data level, whose predictive means often correspond to the standard operator learning models. By leveraging the linearity within the GP framework, DeepOGP offers a total uncertainty that combines the data uncertainty and the function uncertainty. Extensive numerical examples, including nonlinear PDEs (Burgers, reaction–diffusion), and climate modeling, demonstrate the performance of DeepOGP, which provides reliable predictions within estimated uncertainty.