The local prediction of fatigue damage within polycrystals in a high-cycle fatigue setting is a long-lasting and challenging task. It requires identifying grains tending to accumulate plastic deformation under cyclic loading. We address this task by transcribing ferritic steel microtexture and damage maps from experiments into a microstructure graph. Here, grains constitute graph nodes connected by edges whenever grains share a common boundary. Fatigue loading causes some grains to develop slip markings, which can evolve into microcracks and lead to failure. This data set enables applying graph neural network variants on the task of binary grain-wise damage classification. The objective is to identify suitable data representations and models with an appropriate inductive bias to learn the underlying damage formation causes. Here, graph convolutional networks yielded the best performance with a balanced accuracy of 0.72 and an F1-score of 0.34, outperforming phenomenological crystal plasticity (+ 68%) and conventional machine learning (+ 17%) models by large margins. Further, we present an interpretability analysis that highlights the grains along with features that are considered important by the graph model for the prediction of fatigue damage initiation, thus demonstrating the potential of such techniques to reveal underlying mechanisms and microstructural driving forces in critical grain ensembles.