This work presents a fully quantum approach to support vector machine (SVM) learning by integrating gate-based quantum kernel methods with quantum annealing-based optimization. We explore the construction of quantum kernels using various feature maps and qubit configurations, evaluating their suitability through Kernel-Target Alignment (KTA). The SVM dual problem is reformulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem, enabling its solution via quantum annealers. Our experiments demonstrate that a high degree of alignment in the kernel and an appropriate regularization parameter lead to competitive performance, with the best model achieving an F1-score of 90%. These results highlight the feasibility of an end-to-end quantum learning pipeline and the potential of hybrid quantum architectures in quantum high-performance computing (QHPC) contexts.