This paper explores the impact of dc-link voltage control (DVC) on the torsional vibration of grid-forming permanent magnet synchronous generator wind turbines (GFM-WTs), where the DVC is adopted by the machine-side converter. This work investigates three widely used DVC strategies, characterized by different physical implications of output of the DVC. First, a reduced-order small-signal model of the GFM-WT is developed. Employing the complex torque coefficients method, the natural frequency and damping ratio of torsional modes for GFM-WTs are derived, which directly show the impacts of different DVCs on torsional modes. This method also allows for an in-depth analysis of torsional dynamics under different control strategies, considering the varying short-circuit ratio (SCR) of ac grid, dc-link capacitances, and droop and inertia coefficients of GFM control. Finally, nonlinear time-domain simulations are carried out to confirm the theoretical findings.