Elliptic curve cryptography (ECC) is considered as a secure communication encryption method due to its high security-per-bit and efficiency on low power devices. Yet, the validation of ECC-based tasks (e.g. signature validation, key agreement) can be a heavy overhead for lightweight sensors and devices. In this paper, we introduce a novel framework, CurveGNN which uses Graph Neural Network (GNN) with elliptic curve-based structures to quickly and heuristically verify cryptography attributes. By converting the ECC point addition relations and scalar multiplications to graph-structured data, CurveGNN can learn signatures (e.g., ECDSA) validity with marginal arithmetic overhead. We detail the graph construction, GNN architecture, and training strategy. We demonstrate via experiments on embedded-class microcontrollers that CurveGNN achieves more than 95% verification accuracy, with average energy saving over 40% for ECC verification compared to ECC verification, respectively. From our results we observed that CurveGNN is a feasible lightweight verifier in Internet of Things and edge-computing environments.