Identifying earth-abundant catalysts for the hydrogen evolution reaction (HER) is a central challenge in scalable green hydrogen production. We present a computational pipeline that couples diffusion-based generative modelling, machine-learning force-field relaxation, and graph neural network (GNN) property prediction to discover bimetallic HER catalysts beyond the structures and stoichiometries covered by existing adsorption databases. A generative model (MatterGen) is fine-tuned on bulk alloy structures labelled with their minimum hydrogen adsorption energy and steered towards adsorption energies near zero. From the generated pool we retain the stable structures that are novel bimetallic alloys: binary compositions built from the training-set metals but at stoichiometries or structures absent from that database. Each candidate's low-index facets are ranked by a machine-learning-force-field (MatterSim) surface-energy screen, and only the most stable terminations are converted into explicit slab models with hydrogen adsorbed at all high-symmetry surface sites and relaxed with the same force field. Two GNN surrogates (MEGNet and ALIGNN), trained on a high-throughput density functional theory (DFT) dataset, then screen the resulting slab candidates. Because the two models encode the local environment through different inductive biases, we retain only the alloys on which both agree, reducing the influence of any single model. This consensus screening, combined with a MatterSim energy-difference triangulation, resulted in a shortlist of near-thermoneutral bimetallic candidates. Eight are validated with plane-wave DFT, and four of them, Pd5Au, Pt5Rh, Cu5Pt3, and Pd13Ru3, bind hydrogen within the thermoneutral window. All eight alloys are dynamically stable by phonon calculations. These four alloys are highlighted as DFT-validated bimetallic HER candidates for experimental follow-up.