Current defenses against graph attacks often rely on certain properties to eliminate structural perturbations by identifying adversarial edges from normal edges. However, this dependence makes defenses vulnerable to adaptive (white-box) attacks from adversaries with the same knowledge. Adversarial training seems to be a feasible way to enhance robustness without reliance on artificially designed properties. However, in this paper, we show that it can lead to models learning incorrect information. To solve this issue, we re-examine graph attacks from the out-of-distribution (OOD) perspective for poisoning and evasion attacks and introduce a novel adversarial training paradigm incorporating OOD detection. This approach strengthens the robustness of Graph Neural Networks (GNNs) without reliance on prior knowledge. To further evaluate adaptive robustness, we develop adaptive attacks against our methods, revealing a trade-off between graph attack efficacy and defensibility. Through extensive experiments over 25,000 perturbed graphs, our method could still maintain good robustness against both adaptive and non-adaptive attacks. The code is provided at https://github.com/likuanppd/GOOD-AT.