The efficiency of the oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) in energy conversion devices is often hindered by sluggish kinetics and high overpotentials. The role of different nitrogen coordination groups including graphitic N (GN), pyridinic N (PdN), and pyrrolic N (PrN) groups in N-doped carbon materials for these processes is still under debate. Using density functional theory (DFT) calculations, we explored graphene structures doped with in-plane GN, PdN, and PrN as cost-effective electrocatalysts for oxygen electrode reactions, respectively. Our results highlight the importance of explicit solvents in accurately describing the binding behaviours of ORR/OER intermediates, contrasting with vacuum modelling which ignores the hydrogen bond formed between the adsorbates and water layer. Utilizing an explicit water layer, PdN-doped graphene is theoretically recognized as a bifunctional electrocatalyst for oxygen electrode catalysis. Moreover, based on the frontier molecular orbital theory, the Highest Occupied Molecular Orbital (HOMO) energy level of active sites is the underlying factor for the strong binding affinity to oxygen-containing intermediates, leading to the improved ORR/OER activity. Our work provides insight into the identification of active sites in N-doped graphene and provides a theoretical guidance for the rational design of effective carbon-based ORR/OER catalysts.