A sophisticated hybrid quantum convolutional neural network (HQCNN) is conceived for handling the pilot assignment task in cell-free massive MIMO systems, while maximizing the total ergodic sum throughput. The existing model-based solutions found in the literature are inefficient and/or computationally demanding. Similarly, conventional deep neural networks may struggle in the face of high-dimensional inputs, require complex architectures, and their convergence is slow due to training numerous hyperparameters. The proposed HQCNN leverages parameterized quantum circuits (PQCs) relying on superposition for enhanced feature extraction. Specifically, we exploit the same PQC across all the convolutional layers for customizing the neural network and for accelerating the convergence. Our numerical results demonstrate that the proposed HQCNN offers a total network throughput close to that of the excessive-complexity exhaustive search and outperforms the state-of-the-art benchmarks.