This paper presents a novel framework that integrates learnable graph convolutions with the geometric multigrid method for solving partial differential equations (PDEs). The discretization of PDEs is first represented as a graph structure, enabling the application of graph convolutions to enhance the multigrid performance. By incorporating graph convolutions into the multigrid components such as smoothing and inter-grid transfer operators, we develop a learnable multigrid that can adaptively optimize its performance based on the underlying problem characteristics. In this framework, the graph convolutions are embedded directly within the multigrid cycle, effectively transforming the entire multigrid solver into a specialized neural network architecture, rather than combining a classical solver with external GNN modules. The learnable multigrid framework is lightweight in terms of parameter count and requires only one or a few-shot training to achieve good performance. Numerical experiments demonstrate the effectiveness of the proposed approach in solving some challenging problems, showing improved convergence rates compared to traditional multigrid methods.