This paper presents a fully quantum approach to support vector machine (SVM) training by integrating gate-based quantum kernel methods and quantum annealing-based optimization. Quantum kernels are constructed using various feature maps and qubit configurations, and their fitness is evaluated via kernel-target alignment (KTA). The SVM dual problem is reformulated as a quadratic unconstrained binary optimization (QUBO) problem and solved using a quantum annealer. Experimental results demonstrate that the high alignment of the kernels and appropriate regularization parameters contribute to competitive performance, with the best-performing model achieving an F1 score of 90%. These results highlight the feasibility of an end-to-end quantum training pipeline and the potential of hybrid quantum architectures in quantum high-performance computing (QHPC) environments.