Accurate prediction of first-ply failure (FPF) in laminated composite structures with stress concentrators remains a challenging and computationally demanding task. High-fidelity numerical methods are required to resolve localized stress fields near geometric discontinuities, while repeated analyses are often needed during design and optimization processes. This work presents a hybrid physics-informed machine learning (PIML) accelerated boundary element framework for efficient first-ply failure prediction in composite laminates. An anisotropic Boundary Element Method (BEM) is employed to accurately compute local stress concentrations around cut-outs, while Classical Laminate Theory (CLT) provides the global laminate response. BEM-generated stress data are then used to train a physics-informed machine learning surrogate model, constrained by mechanical consistency and material symmetries. The surrogate model enables rapid evaluation of stress concentration effects under varying ply orientations and loading conditions without replacing the underlying physics-based formulation. First-ply failure is assessed using established composite failure criteria, ensuring full compatibility with classical composite mechanics. The proposed framework significantly reduces computational cost compared to repeated full BEM simulations while maintaining the accuracy required for reliable failure prediction. The results demonstrate that physics-informed machine learning can effectively enhance boundary-based numerical methods, providing a robust and efficient tool for composite laminate design and structural integrity assessment.