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In Zhu and Qiu (J Comput Phys 228:6957-6976, ), we systematically investigated adaptive Runge-Kutta discontinuous Galerkin (RKDG) methods for hyperbolic conservation laws with different indicators, with an objective of obtaining efficient and reliable indicators to obtain better performance for adaptive computation to save computational cost. In this follow-up paper, we extend the method to solve two-dimensional problems. Although the main idea of the method for two-dimensional case is similar to that for one-dimensional case, the extension of the implementation of the method to two-dimensional case is nontrivial because of the complexity of the adaptive mesh with hanging nodes. We lay our emphasis on the implementation details including adaptive procedure, solution projection, solution reconstruction and troubled-cell indicator. Extensive numerical experiments are presented to show the effectiveness of the method. [ABSTRACT FROM AUTHOR]