The distance of a quantum code is a fundamental metric that dictates its error-correction capability. While quantum low-density parity-check (qLDPC) families such as Bivariate Bicycle (BB) codes offer a promising pathway to low-overhead fault-tolerant quantum computing, computing their distance is challenging. Current exact solvers and even search heuristics that provide a distance upper bound suffer from severe computational bottlenecks, restricting the discovery of optimal codes. In this paper, we construct a dataset of BB codes with corresponding distance upper bounds obtained through high-budget heuristic searches. We introduce BB-Net, an attention-based neural network architecture for predicting the distance of BB codes. It achieves 96.39% accuracy on the reference distances in the test set, compared with 94.86% for state-of-the-art heuristics with a budget of up to $10^{4}$ times more computational runtime. On a subset with certified exact distances, BB-Net achieves 98.94%. By relieving the distance-calculation bottleneck, BB-Net enables faster screening of large code spaces, particularly in ML-driven search pipelines, and thereby can accelerate the discovery of novel quantum error-correcting codes, an important capability for advancing scalable fault-tolerant quantum computing. Our code is publicly available: https://github.com/NadavSc/BBNet

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