The Monte Carlo method, which is based on finite element models directly, is extremely time-consuming for quantifying the uncertainty of structural dynamic characteristics. To address the above issue, Gaussian process model was introduced to replace the time-intensive finite element model to enhance the computational efficiency of uncertainty quantification. A method for uncertainty quantification of structural dynamic characteristics was proposed based on sequential design and Gaussian process models. Optimal sample points were selected to establish an adaptive Gaussian process model through iterative sample enrichment criteria, thereby improving the accuracy of uncertainty quantification. The high-dimensional integration of statistical moments of dynamic characteristics was transformed into one-dimensional integration under the framework of the established adaptive Gaussian process model, allowing for analytical computation. Two mathematical functions were used to illustrate the fitting process of the adaptive Gaussian model, indicating a noticeable increase of the fitting accuracy with the increase of the number of iterations. Subsequently, the proposed method was applied to the calculation of the statistical moments of natural frequencies for a cylindrical shell, with computational accuracy comparable to that of the Monte Carlo method. The proposed method demonstrated significant advantage in computational accuracy and efficiency, in comparison with the traditional Gaussian process models.