Deep unfolding networks based on total variation (TV) regularization provide an interpretable framework for image restoration by integrating model-driven optimization with data-driven learning. However, existing primal-dual unfolding networks often require a large number of iterative stages to achieve satisfactory restoration quality, leading to increased computational cost and slow convergence. To address this limitation, we propose a fast trainable primal-dual TV unfolding network with a learnable momentum acceleration mechanism. Building upon the classical primal-dual hybrid gradient optimization framework, an adaptive momentum strategy is introduced to accelerate information propagation across unfolding stages. Specifically, a dual-mechanism design combining exponential momentum decay and temporal fusion is developed to balance convergence speed and numerical stability. The resulting accelerated unfolding architecture retains the interpretability of variational optimization while significantly improving training and inference efficiency. Extensive experiments on image denoising benchmarks demonstrate that the proposed framework consistently achieves superior restoration performance with fewer unfolding stages and reduced runtime compared with existing TV-based and learning-based methods. The results verify the effectiveness of learnable momentum acceleration for efficient variational image restoration.