We conducted ab initio molecular dynamics simulations to investigate the dynamic properties of the liquid Ti-Al-Ni ternary alloy system at 2033 K. Through simulations across the entire composition range, we revealed the complex compositional dependence of self-diffusion coefficients and viscosity. The self-diffusion coefficients of Ti, Al, and Ni exhibit nonlinear distribution characteristics, while viscosity shows non-monotonic compositional dependence, with lower viscosity in Al-rich regions and higher viscosity in Ni-rich regions. To understand the structural origins of these behaviors, we analyzed chemical short-range order and local fivefold symmetry structures using Warren-Cowley parameters and fivefold symmetry parameters, uncovering a close correlation between atomic-scale structural features and dynamic properties. Evaluation of the Stokes-Einstein relation revealed varying degrees of deviation from its predicted viscosity values across composition regions. We observed a negative correlation between free volume fraction and viscosity, with varying strengths in different regions. These findings provide new perspectives for understanding the complex dynamic properties of Ti-Al-Ni liquid alloys.