In recent years, progress in adaptive graph signal processing algorithms has provided effective solutions for processing signals defined on graph structures. As a classical strategy in information theory, the Generalized Maximum Correntropy Criterion (GMCC) exhibits good resistance to non-Gaussian noises. When non-Gaussian noise interferes with the graph signal, the graph signal processing algorithm based on GMCC (GSP GMCC) algorithm shows better performance. However, the GSP GMCC algorithm itself has three parameters that need to be manually tuned, and the process of manually tuning the parameters is complex and tedious. Meanwhile, the non-concave and non-convex nature of the GMCC function itself limits its own convergence rate and adaptive estimation accuracy. To solve the above problems, based on the strongly convex function half-quadratic criterion (HQC), the GSP HQC algorithm is proposed in this paper. The performance analysis of the GSP HQC algorithm is implemented in this paper. Simulation experiments demonstrate that the GSP HQC algorithm achieves superior performance in terms of convergence rate and adaptive estimation accuracy while maintaining computational complexity comparable to existing algorithms