We propose a novel algorithm for estimating Gaussian graphical models incorporating prior information about the underlying graph. Classical approaches generally propose optimization problems with sparsity penalties as prior information. While efficient, these approaches do not allow using involved prior distributions and force us to incorporate the prior information on the precision matrix rather than on its support. In this work, we investigate how to estimate the graph of a Gaussian graphical model by introducing any prior distribution directly on the graph structure. We use graph neural networks to learn the score function of any graph prior and then leverage Langevin diffusion to generate samples from the posterior distribution. We study the estimation of both partially known and entirely unknown graphical models and prove that our proposed estimator is consistent in both scenarios. Finally, numerical experiments using synthetic and real-world graphs demonstrate the benefits of our approach.