Speech intelligibility and quality assessment models are essential tools for researchers to evaluate and improve speech processing models. However, only a few studies have investigated multi-task models for intelligibility and quality assessment due to the limitations of available data. In this study, we released TMHINT-QI, the first Chinese speech dataset that records the quality and intelligibility scores of clean, noisy, and enhanced utterances. Then, we propose InQSS, a non-intrusive multi-task learning framework for intelligibility and quality assessment. We evaluated the InQSS on both the training-from-scratch and the pretrained models. The experimental results confirm the effectiveness of the InQSS framework. In addition, the resulting model can predict not only the intelligibility scores but also the quality scores of a speech signal.