With the proliferation of large-scale grid-connected wind farms, subsynchronous oscillation (SSO) incidents associated with Type-4 wind turbines (WTs) with a permanent magnet synchronous generator (PMSG) have occurred frequently. These incidents have caused severe reliability risks to the power grid. Conventionally, P-Q capability charts are utilized to ensure the safety operating region of a synchronous generator. However, a PMSG WT exhibits completely different and dynamic P-Q capability characteristics due to the difference in energy conversion technique and several critical factors related to the WT power converters. This paper presents a comprehensive dynamic P-Q capability study of a PMSG WT with sufficient and accurate considerations of the WT control and operation in the dq reference frame, its power converter constraints, and grid dynamics. Models of a PMSG WT are first developed based on its control principle in the dq reference frame. Then, algorithms for obtaining the P-Q capability charts of the WT are developed with the considerations of complete WT constraints in different aspects. The study analyzes the root cause of many abnormal operations of grid-connected PMSG WTs, reported in the literature, from the dynamic P-Q capability perspectives. The proposed study is verified via an electromagnetic transient (EMT) model of a grid-connected Type-4 WT.