The present paper proposes an efficient inverse design framework that integrates Bayesian optimization and Bayesian inversion to enable accurate parameter estimation and the rapid construction of surrogate models, while ensuring rigorous uncertainty quantification. The framework employs Bayesian optimization to adaptively construct Gaussian process surrogate models, efficiently exploring the parameter space with a limited number of high-fidelity model evaluations. Once the surrogate model is sufficiently trained, Bayesian inversion is performed to estimate the most probable model parameters by combining prior knowledge with observed data. This process yields posterior probability distributions that characterize the epistemic uncertainty associated with the inferred parameters. The synergy between the two Bayesian approaches offers significant advantages: Bayesian optimization accelerates surrogate modeling by focusing sampling efforts in regions of high predictive uncertainty, while Bayesian inversion refines parameter estimates within a probabilistic framework that may further reduce the feasible parameter domain. This integrated strategy not only enhances computational efficiency but also provides a systematic and comprehensive treatment of uncertainty in inverse design problems. The effectiveness of the proposed methodology is demonstrated through a set of analytical benchmark functions in one and two dimensions, including the Mixed Gaussian-Periodic, Lévy, Griewank, and Forrester functions. Results show that the combined Bayesian framework enables the fast construction of accurate surrogate models and delivers informative uncertainty quantification in complex inverse scenarios.