Spike-timing-dependent synaptic plasticity (STDP) is demonstrated in a synapse device based on a ferroelectric-gate field-effect transistor (FeFET). STDP is a key of the learning functions observed in human brains, where the synaptic weight changes only depending on the spike timing of the pre- and post-neurons. The FeFET is composed of the stacked oxide materials with ZnO/Pr(Zr,Ti)O3 (PZT)/SrRuO3. In the FeFET, the channel conductance can be altered depending on the density of electrons induced by the polarization of PZT film, which can be controlled by applying the gate voltage in a non-volatile manner. Applying a pulse gate voltage enables the multi-valued modulation of the conductance, which is expected to be caused by a change in PZT polarization. This variation depends on the height and the duration of the pulse gate voltage. Utilizing these characteristics, symmetric and asymmetric STDP learning functions are successfully implemented in the FeFET-based synapse device by applying the non-linear pulse gate voltage generated from a set of two pulses in a sampling circuit, in which the two pulses correspond to the spikes from the pre- and post-neurons. The three-terminal structure of the synapse device enables the concurrent learning, in which the weight update can be performed without canceling signal transmission among neurons, while the neural networks using the previously reported two-terminal synapse devices need to stop signal transmission for learning.