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The Geoscience Laser Altimeter System (GLAS) aboard Ice, Cloud and land Elevation Satellite (ICESat) was able to capture the full waveform of backscattered laser pulse. However, the accuracy of the surface information extracted from the waveform was vulnerable to background noise. In this paper, a piecewise adaptive l_q-norm trend filtering method is proposed for the GLAS full waveform denoising on the basis of trend filtering. To minimize the loss of useful signal while removing the noise, the proposed method adaptively assigns different norms to the smooth constraints according to the local signal energy. The filtered results can then be obtained by iteratively minimizing the hybrid-norm loss function. The proposed method is tested on both the simulated waveforms and real GLAS waveform data. In the simulated experiments, the quantitative evaluation is conducted with the filtered waveforms, as well as the results after waveform decomposition. For comparison, the most commonly used waveform filtering methods, i.e. Gaussian filtering, wavelet transform, Empirical model decomposition and l₁ trend filtering, are involved in the experiments. The results show that the proposed method outperforms the mainstream methods on waveform filtering, in terms of removing noise and preserving the shape and energy amplitude of the GLAS waveforms.