Quality assessment (QA) and brain extraction (BE) are two fundamental steps in 3D fetal brain MRI reconstruction and quantification. Conventionally, QA and BE are performed independently, ignoring the inherent relation of the two closely-related tasks. However, both of them focus on the brain region representation, so they can be jointly optimized to ensure the network to learn shared features and avoid overfitting. To this end, we propose a novel multi-stage deep learning model for joint QA and BE of fetal MRI. The locations and orientations of fetal brains are randomly variable, and the shapes and appearances of fetal brains change remarkably across gestational ages, thus imposing great challenges to extract shared features of QA and BE. To address these problems, we firstly design a brain detector to locate the brain region. Then we introduce the deformable convolution to adaptively adjust the receptive field for dealing with variable brain shapes. Finally, a task-specific module is used for image QA and BE simultaneously. To obtain a well-trained model, we further propose a multi-step training strategy. We cross validate our method on two independent fetal MRI datasets acquired from different scanners with different imaging protocols, and achieve promising performance.