In electrical engineering, solving low-frequency Maxwell's equations is crucial for modeling devices with ferromagnetic materials and large air volumes, typically due to voids and gaps. The Magnetostatic Moment Method (MMM), based on volume integral equations, offers an efficient alternative to the Finite Element Method (FEM) in such cases. The MMM's advantage lies in not requiring the meshing of air regions, significantly reducing the number of unknowns. However, the MMM results in linear systems with dense matrices, leading to computational challenges in terms of memory and execution time for large problems. To address this issue, we consider a low-rank approximation of the matrix using the Fast Multipole Method (FMM). This hierarchical algorithm, which uses a tree-based domain partitioning, can reduce the complexity of matrix-vector product calculations from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$. Our work focuses on adapting a black-box FMM variant, available in the C++ ScalFMM library, to accelerate matrix-vector products within the MMM framework. Additionally, we developed a parallel, shared memory-based version to further enhance computational efficiency. Our numerical results demonstrate the effectiveness of this enhanced FMM-based matrix-vector product in a simplified application modeling a low-frequency antenna.