As a promising physical layer security technique, physical layer key generation (PLKG) enables legitimate users to obtain secret keys from wireless channel without security infrastructures. However, in harsh propagation environments, the channel characteristic becomes unsatisfactory, the key generation rate (KGR) is significantly deteriorated. In this paper, we propose a novel fluid antenna (FA) enabled PLKG system to address this challenge. Specifically, we first derive the closed-form expression of the KGR for FA array, and then jointly optimize the precoding matrix and the antenna positions via a particle swarm optimization (PSO) algorithm. Next, to further reduce the computational complexity of the optimization procedure, we develop an alternating optimization (AO) algorithm, which combines the projected gradient descent (PGD) and the PSO. Simulation results demonstrate that by exploiting the additional spatial degree of freedom (DoF), our FA enabled PLKG system is superior to the benchmarks, such as the conventional fixed-position antenna (FPA) array and the reconfigurable intelligent surface (RIS). It is worth highlighting that compared to the conventional uniform planar antenna (UPA), the FA enabled PLKG achieves a 35.42\% KGR performance improvement under PSO algorithm and a 67.73\% KGR performance improvement under AO algorithm, respectively.