A popular machine-learning model for regression tasks, including stock-market prediction, weather forecasting and real-estate pricing, is the classical support vector regression (SVR). However, a practically realisable quantum SVR remains to be formulated. We devise annealing-based algorithms, namely simulated and quantum-classical hybrid, for training two SVR models and compare their empirical performances against the SVR implementation of Python's scikit-learn package for facial-landmark detection (FLD), a particular use case for SVR. Our method is to derive a quadratic-unconstrained-binary formulation for the optimisation problem used for training a SVR model and solve this problem using annealing. Using D-Wave's hybrid solver, we construct a quantum-assisted SVR model, thereby demonstrating a slight advantage over classical models regarding FLD accuracy. Furthermore, we observe that annealing-based SVR models predict landmarks with lower variances compared to the SVR models trained by gradient-based methods. Our work is a proof-of-concept example for applying quantum-assisted SVR to a supervised-learning task with a small training dataset.