Simultaneous multi-index quantification, segmentation, and uncertainty estimation of liver tumors on multi-modality non-contrast magnetic resonance imaging (NCMRI) are crucial for accurate diagnosis. However, existing methods lack an effective mechanism for multi-modality NCMRI fusion and accurate boundary information capture, making these tasks challenging. To address these issues, this paper proposes a unified framework, namely edge-aware multi-task network (EaMtNet), to associate multi-index quantification, segmentation, and uncertainty of liver tumors on the multi-modality NCMRI. The EaMtNet employs two parallel CNN encoders and the Sobel filters to extract local features and edge maps, respectively. The newly designed edge-aware feature aggregation module (EaFA) is used for feature fusion and selection, making the network edge-aware by capturing long-range dependency between feature and edge maps. Multi-tasking leverages prediction discrepancy to estimate uncertainty and improve segmentation and quantification performance. Extensive experiments are performed on multi-modality NCMRI with 250 clinical subjects. The proposed model outperforms the state-of-the-art by a large margin, achieving a dice similarity coefficient of 90.01$\pm$1.23 and a mean absolute error of 2.72$\pm$0.58 mm for MD. The results demonstrate the potential of EaMtNet as a reliable clinical-aided tool for medical image analysis.