Elastic Optical Network (EON) is a promising solution to address the high capacity, low latency, and flexibility requirements of the upcoming 5th-generation (5G) networks. Furthermore, Multi-Core Fibers (MCFs) and Space Division Multiplexing (SDM) technique can be utilized to overcome the capacity limitation of the conventional Single Mode Fibers (SMFs). On the other hand, Quantum Key Distribution (QKD) is an effective solution to address the security issues in 5G transport networks. In this paper, we investigate the performance of QKD over elastic optical networks with multi-core fibers and address the resource allocation problem for quantum and classical channels of QKD (QChs and CChs) and conventional data channels (DChs). To do so, we calculate the background noise caused by different noise sources and accordingly calculate the Secret Key Rate (SKR) in quantum channels. Then, we propose an Integer Linear Programming formulation and a heuristic algorithm to allocate network resources (spectrum, core, and links) to QChs, CChs, and DChs, with the objective of maximizing the secret key rate and minimizing the number of utilized frequency slots (FSs). Finally, we evaluate the proposed ILP and heuristic algorithm in terms of SKR and the number of utilized FSs. In our simulations, we consider core and metro topologies, fixed and distance adaptive launch power for classical signals, different fiber specifications, and different assumptions regarding the relative locations of quantum and classical channels in a multi-core fiber.