•A complete ABQ-based procedure for first-four moments estimation of PDF.•An efficient reliability method in conjunction with ABQ-based procedure and SGLD model.•Adaptive sampling scheme formulated to iteratively add new integration points.•Goodness of moments estimation guaranteed by prescribing a stopping condition.•The proposed method superior to existing ‘non-adaptive’ integration methods. An efficient method for structural reliability analysis is presented, which combines the statistical moments estimation and a versatile probability distribution model, i.e., the shifted generalized lognormal distribution (SGLD) model. In this method, a complete adaptive Bayesian quadrature (ABQ)-based procedure is developed to evaluate the first-four central moments of the equivalent extreme value (EEV) of structural responses, where an adaptive sampling scheme is formulated to iteratively add new integration points so as to satisfy the prescribed accuracy requirement with as fewer deterministic response analyses as possible. The probability density function (PDF) of the EEV is then recovered by fitting the SGLD model using the estimated statistical moments. Finally, the structural reliability is obtained by a simple one-dimensional integral of the PDF of the EEV over the safe domain. To demonstrate the efficacy of the proposed methodology, two numerical examples are carried out, involving the statistical moments estimation of analytical functions with different mathematical characteristics and the dynamic reliability assessment of nonlinear stochastic structures subjected to seismic excitations. It is revealed that the proposed method is capable of attaining fairly desired reliability measures on engineering structures with satisfactory accuracy and efficiency.