Motivated by loop quantum gravity, we propose two polymer quantization schemes to describe the propagation of gravitational waves within a classical Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime. These novel schemes yield modified waveforms, altering the amplitude of the waves as they traverse quantum spacetime. Additionally, the speed of the waves becomes frequency-dependent due to polymer corrections. We investigate the detectability of these signals using instruments like LISA and LIGO, establishing bounds on the polymer scales.