Despite its superior soft tissue contrast, conventional MRI is qualitative in nature and this presents a bottleneck problem in quantitative image analysis and data-driven medicine. Various investigations have been devoted to overcoming this limitation, but practical solutions remain elusive. Leveraging from the unique ability of emerging deep learning in feature extraction, we investigate a data-driven strategy of separating contributions of various contributing factors intertwined in a single T1 weighted image to derive quantitative T1 and ρ maps without any additional image acquisition. Furthermore, in the proposed deep learning framework, compensation for radiofrequency field inhomogeneities is automatically achieved without extra measurement of B1 map. The tasks are accomplished using self-attention deep convolutional neural networks, which make efficient use of both local and non-local information. The premise of the approach is that qualitative and quantitative MRI, named Q2MRI, can be attained simultaneously without changing the existing imaging protocol. Q2MRI lays foundation for next generation of digital medicine and provides a promising quantitative imaging tool for a wide spectrum of biomedical applications, ranging from disease diagnosis, treatment planning, prognosis to assessment of therapeutic response.