The offshore-onshore integrated energy system (OOIES) comprises offshore gas production platforms, wind farms, and onshore gas-fired combined heat and power plants, facilitating the integrated operation of multiple energy sources. To address the challenge of optimally configuring the device capacities in carbon capture and power to gas (CC-P2G) amid stochastic fluctuations in offshore gas and wind power outputs, this study proposes a multi-objective approximate dynamic programming algorithm. This algorithm solves the multi-objective stochastic optimal configuration for the device capacities in CC-P2G in OOIES by simultaneously optimizing investment and operation costs, wind power curtailment, and carbon emissions. By leveraging value function matrices for multiple objectives to solve the extended Bellman equation, the multi-objective multi-period model is decomposed into a series of multi-objective single-period optimization problems, which are solved recursively. Additionally, a weighted Chebyshev function is introduced to obtain the compromise optimal solution for multi-objective optimization model during each period. A case study of an OOIES confirms the effectiveness and efficiency of the proposed algorithm.