Traditional Physics-Informed Distributed Federated Learning (PIDFL) for solving the 2D Poisson equation faces challenges of slow convergence and high communication costs. We propose a Krylov-accelerated federated PINN framework that integrates matrix-free Kronecker Factored Approximate Curvature (K-FAC) with Conjugate Gradient (CG), exploiting the symmetric positive definite structure of discretized Poisson problems to enable scalable second-order optimization in distributed settings. Compared to first-order methods like Adam, our approach reduces communication rounds and achieves improved accuracy. Results across multiple grid sizes demonstrate faster convergence and superior physics residual minimization, supporting its applicability to PDE-constrained learning with communication constraints.