Explaining graphs and their target Graph Neural Networks (GNNs) has gained attention with the growing use of GNNs. Most existing explainable AI (XAI) methods for GNNs focus on extracting an explanation subgraph and assume the target GNN is supervised with accessible class probabilities. However, the growing prevalence of GNN models in unsupervised settings underscores the necessity for task-irrelevant explanations. Moreover, most existing studies scarcely explore whether identifying edges absent from the original graph can improve explanation quality. To this end, we propose HINT-G (Harnessing INfluence function for Task-irrelevant explanation on Graph neural networks), a method that uses influence functions to explain models across diverse learning paradigms and considers edges beyond the given graph. The influence of an edge can be determined directly or by aggregating the influence scores of its constituent nodes, while the influence of a non-existent edge can also be determined. Furthermore, this method is task-irrelevant, since the influence score can be obtained whenever the loss function of the target model is differentiable. Experimental results on several datasets consistently demonstrate that HINT-G effectively explains graphs through the influence function framework. Our implementation code is available at https://github.com/cycy-kim/HINT-G.