We present a hybrid quantum-classical framework augmented with learned error mitigation for solving the viscous Burgers equation on noisy intermediate-scale quantum (NISQ) hardware. Using the Cole-Hopf transformation, the nonlinear Burgers equation is mapped to a diffusion equation, discretized on uniform grids, and encoded into a quantum state whose time evolution is approximated via Trotterized nearest-neighbor circuits implemented in Qiskit. Quantum simulations are executed on noisy Aer backends and IBM superconducting quantum devices and are benchmarked against high-accuracy classical solutions obtained using a Krylov-based solver applied to the corresponding discretized Hamiltonian. From measured quantum amplitudes, we reconstruct the velocity field and evaluate physical and numerical diagnostics, including the L2 error, shock location, and dissipation rate, both with and without zero-noise extrapolation (ZNE). To enable data-driven error mitigation, we construct a large parametric dataset by sweeping viscosity, time step, grid resolution, and boundary conditions, producing matched tuples of noisy, ZNE-corrected, hardware, and classical solutions together with detailed circuit metadata. Leveraging this dataset, we train an attention-based graph neural network that incorporates circuit structure, light-cone information, global circuit parameters, and noisy quantum outputs to predict error-mitigated solutions. Across a wide range of parameters, the learned model consistently reduces the discrepancy between quantum and classical solutions beyond what is achieved by ZNE alone. We discuss extensions of this approach to higher-dimensional Burgers systems and more general quantum partial differential equation solvers, highlighting learned error mitigation as a promising complement to physics-based noise reduction techniques on NISQ devices.