<p>The increasing application of fiber-reinforced polymer (FRP) composite materials to advanced engineering structures necessitates trustworthy and economic calculation procedures for the characterization of their complex mechanical response. Here, the materials are distinguished by strong directional anisotropy due to fiber reinforcement as well as viscoelasticity of the polymer matrix, both of which lead to dynamically growing stress sensitivity under time-dependent or dynamic cyclic loading. Traditional modeling approaches, being either generally isotropic or adopting simplified viscoelastic approximations, lack the ability to capture these coupled effects entirely, especially when fractional viscoelastic effects are dominant. To bridge these challenges, this paper presents a new time-stepping fractional Convolution Quadrature Boundary Element Method (CQBEM) unifying boundary-only discretization, fractional calculus, and convolution quadrature in an integrated framework. The method properly captures the dynamic stress sensitivity of anisotropic viscoelastic FRP composite materials by incorporating direction-dependent elasticity, memory-dependent strain, and reinforcement heterogeneity. Consistency with finite element models and experiments confirms the method’s strength, accuracy, and computational efficiency. Parametric analyses illustrate that stress sensitivity is significantly influenced by fiber direction, retardation times, fractional derivative order, and composite type (CFRP, GFRP, AFRP). This formulation establishes CQBEM as a strong performance predictor of long-term reliability and performance in FRP components, with ready direct application in aerospace, civil, and automotive engineering.</p>