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In this work, we explore how modified gravity theories based on the non-metricity scalar, known as $f(Q)$ gravity, affect the propagation of gravitational waves from inspiraling of binary systems. We discuss forecast constraints on $f(Q)$ gravity by considering standard siren events in two contexts: i) simulated sources of gravitational waves as black hole - neutron star binary systems, emitting in the frequency band of the third-generation detector represented by the Einstein Telescope (ET); ii) three standard siren mock catalogs based on the merger of massive black hole binaries that are expected to be observed in the operating frequency band of the Laser Interferometer Space Antenna (LISA). We find that, within the ET sensitivity, in combination with supernova and cosmic chronometer data, it will be possible to test deviations from general relativity at $<3\%$ accuracy in the redshift range $0<z<5$, while the main free parameter of the theory is globally constrained at 1.6\% accuracy within the same range. In light of LISA's forecasts, combined with supernova and cosmic chronometer data, in the best scenario, we find that the main free parameter of the theory will be constrained at 1.6\% accuracy up to high redshifts. Therefore, we conclude that future gravitational wave observations by ET and LISA will provide a unique way to test, with good accuracy, the nature of gravity up to very large cosmic distances.