This work proposes an adaptive quantum approximate optimization-based model predictive control (MPC) strategy for energy management in buildings equipped with battery energy storage and renewable energy generation systems. The learning-based parameter transfer scheme to realize adaptive quantum optimization leverages Bayesian optimization to predict initial quantum circuit parameters. When applied to the MPC problems formulated as quadratic unconstrained binary optimization problems, this approach computes optimal controls to minimize the net energy consumption levels in buildings and promotes decarbonization while reducing the computational efforts required for the quantum approximate optimization algorithm as the building energy system trajectory progresses. The energy efficiency and the decarbonization benefits of the proposed quantum optimization-based MPC strategy are demonstrated on buildings at the Cornell University campus. The proposed quantum computing-based technique to address MPC problems in buildings demonstrates energy-efficient and low-carbon building operation with a 6.8% improvement over deterministic MPC and presents opportunities for scaling to larger control problems with a significant reduction in utilized quantum computing resources. A reduction of 41.2% in carbon emissions is also achieved with the proposed control strategy facilitated by efficiently managing battery energy storage and renewable generation sources to promote a push toward carbon-neutral building operations.