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Quantum entanglement as the one of the most general quantum resources, can be quantified by von Neumann entropy. However, as we know, the von Neumann entropy is only statistical quantity or operator, it therefore has fluctuation. The quantum fluctuation of entanglement (QFE) between Unruh-Dewitt detector modeled by a two-level atom is investigated in a relativistic setting. The Unruh radiation and quantum fluctuation effects affect the precise measurement of quantum entanglement. Inspired by this we present how the relativistic motion effects QFE for two entangled Unruh-Dewitt detectors when one of them is accelerated and interacts with the neighbor external scalar field. We find that QFE first increases by the Unruh thermal noise and then suddenly decays when the acceleration reaches at a considerably large value, which indicates that relativistic effect will lead to non-negligible QFE effect. We also find that the initial QFE (without acceleration effect) is minimum with the maximally entangled state. Moreover, although QFE has a huge decay when the acceleration is greater than $\sim0.96$, concurrence also decays to a very low value, the ratio $ΔE/C$ therefore still large. According to the equivalence principle, our findings could be in principle applied to dynamics of QFE under the influence of gravitation field.