K-nearest neighbor search is one of the fundamental tasks in various applications and the hierarchical navigable small world (HNSW) has recently drawn attention in large-scale cloud services, as it easily scales up the database while offering fast search. On the other hand, a computational storage device (CSD) that combines programmable logic and storage modules on a single board becomes popular to address the data bandwidth bottleneck of modern computing systems. In this paper, we propose a computational storage platform that can accelerate a large-scale graph-based nearest neighbor search algorithm based on SmartSSD CSD. To this end, we modify the algorithm more amenable on the hardware and implement two types of accelerators using HLS- and RTL-based methodology with various optimization methods. In addition, we scale up the proposed platform to have 4 SmartSSDs and apply graph parallelism to boost the system performance further. As a result, the proposed computational storage platform achieves 75.59 query per second throughput for the SIFT1B dataset at 258.66W power dissipation, which is 12.83x and 17.91x faster and 10.43x and 24.33x more energy efficient than the conventional CPU-based and GPU-based server platform, respectively. With multi-terabyte storage and custom acceleration capability, we believe that the proposed computational storage platform is a promising solution for cost-sensitive cloud datacenters.