In cardiac Magnetic Resonance Imaging (MRI) analysis, T2 mapping is a tissue quantification technology to measure water and inflammation levels and has been recognized as an important versatile index for myocardial pathologies. The T2 quantification is a highly relevant task with the myocardial segmentation but they only have been explored separately in deep learning. Here, we proposed a simultaneous dual-task network for myocardial segmentation and T2 quantification, called SQNet. It integrates Transformer and Convolutional Neural Network (CNN) components. SQNet has a T2-refine fusion decoder for quantitative analysis, incorporating global features extracted by the Transformer, and employs a segmentation decoder with multiple local region supervision to enhance segmentation accuracy. Additionally, SQNet proposes a tight coupling module that aligns and fuses CNN and Transformer branch features. The mutual promotion of the two tasks enables the SQNet to focus on the myocardium regions. The dual-task performance is validated on healthy controls (HC, N=17) and acute myocardial infarction patients (AMI, N=12). The segmentation dice of SQNet on HC/AMI is 89.3/89.2, which is higher than that (87.7/87.9) of the compared state-of-the-art segmentation method. The Pearson correlation coefficient of the T2 value on HC/AMI is 0.84 and 0.93 to the label value, indicating a strong linear correlation in myocardial T2 mapping. From the clinical diagnostic perspective, eight experienced radiologists have evaluated the image quality (Score standard: Good is 4.0 and excellent is 5.0) of SQNet on HC/AMI with the segmentation score 4.60/4.58 and the T2 quantification score 4.32/4.42, which is higher than the compared state-of-the-art segmentation score (4.50/4.44) and the T2 quantification score (3.59/4.37). Thus, SQNet provides an accurate simultaneous segmentation and quantification, providing the more accurate diagnosis of cardiac disease, such as AMI.