Speech quality assessment has been a critical component in many voice communication related applications such as telephony and online conferencing. Traditional intrusive speech quality assessment requires the clean reference of the degraded utterance to provide an accurate quality measurement. This requirement limits the usability of these methods in real-world scenarios. On the other hand, non-intrusive subjective measurement is the ``golden standard" in evaluating speech quality as human listeners can intrinsically evaluate the quality of any degraded speech with ease. In this paper, we propose a novel end-to-end model structure called Convolutional Context-Aware Transformer (CCAT) network to predict the mean opinion score (MOS) of human raters. We evaluate our model on three MOS-annotated datasets spanning multiple languages and distortion types and submit our results to the ConferencingSpeech 2022 Challenge. Our experiments show that CCAT provides promising MOS predictions compared to current state-of-art non-intrusive speech assessment models with average Pearson correlation coefficient (PCC) increasing from 0.530 to 0.697 and average RMSE decreasing from 0.768 to 0.570 compared to the baseline model on the challenge evaluation test set.