Quantum key distribution (QKD) holds the potential of providing long-term integrity and confidentiality for data and communications. Currently, many fiber-based QKD systems have been commercialized and several QKD networks have been deployed. Given the high cost and complexity of QKD network deployment, QKD as a service (QaaS) becomes a promising pattern for future QKD networks. The QaaS concept is that multiple users can apply for QKD services to obtain their required secret-key rates (SKRs) from the same QKD network infrastructure instead of deploying their dedicated QKD networks. Accordingly, how to provide efficient and flexible QaaS for fulfilling the SKR requirements of multiple users over a QKD network infrastructure becomes a new challenge. This study introduces the software defined networking (SDN) technique to overcome this challenge, since SDN can add flexibility together with efficient QKD network management. A new framework of SDN for QaaS (SDQaaS) is proposed, where the QaaS functions are developed in the SDN controller. We present the protocol extension, intercommunication workflow, and routing and SKR assignment strategy for QaaS implementation in the SDQaaS framework. We also establish a SDQaaS experimental testbed and perform the numerical simulation to verify our presented approaches. Experimental results demonstrate that our presented approaches can achieve efficient and flexible QaaS over the QKD network. Moreover, simulation results indicate that the success probability of QKD service requests can be increased via lowering the flexibility of SKR requirements for QKD service creation, sacrificing more cost to produce higher SKR over the QKD network, or gradually reducing SKR requirements with the modification of QKD service.