We proposed an end-to-end deep learning convolutional neural network (DCNN) for region-of-interest based multi-parameter quantification (RMQ-Net) to accelerate quantitative ultrashort echo time (UTE) MRI of the knee joint with automatic multi-tissue segmentation and relaxometry mapping. The study involved UTE-based T1 (UTE-T1) and Adiabatic T1ρ (UTE-AdiabT1ρ) mapping of the knee joint of 65 human subjects, including 20 normal controls, 29 with doubtful-minimal osteoarthritis (OA), and 16 with moderate-severe OA. Comparison studies were performed on UTE-T1 and UTE-AdiabT1ρ measurements using 100%, 43%, 26%, and 18% UTE MRI data as the inputs and the effects on the prediction quality of the RMQ-Net. The RMQ-net was modified and retrained accordingly with different combinations of inputs. Both ROI-based and voxel-based Pearson correlation analyses were performed. High Pearson correlation coefficients were achieved between the RMQ-Net predicted UTE-T1 and UTE-AdiabT1ρ results and the ground truth for segmented cartilage with acceleration factors ranging from 2.3 to 5.7. With an acceleration factor of 5.7, the Pearson r-value achieved 0.908 (ROI-based) and 0.945 (voxel-based) for UTE-T1, and 0.733 (ROI-based) and 0.895 (voxel-based) for UTE-AdiabT1ρ, correspondingly. The results demonstrated that RMQ-net can significantly accelerate quantitative UTE imaging with automated segmentation of articular cartilage in the knee joint.