In this study, gravitational waveforms emitted by inspiralling compact binary systems on quasicircular orbits in hybrid metric-Palatini gravity are computed in the lowest post-Newtonian approximation. By applying the stationary phase approximation, Fourier transforms of the tensor polarization modes are obtained, and correction terms in the amplitude and phase of gravitational waves relative to General Relativity results are derived. Moreover, post-Einsteinian parameters are identified, and potential constraints on the background value of the scalar field are obtained based on possible observations of the gravitational waves by the future ground-based gravitational wave detectors. Additionally, constraints on the background value of the scalar field are derived using updated observational data from the PSR J0737-3039 system. These latter constraints are comparable in order of magnitude to the best currently existing constraints, which were derived from observational data within the Solar System.