Single tracking location (STL) shear wave elasticity imaging (SWEI) is a method for detecting elastic differences between tissues. It has the advantage of intrinsic speckle bias suppression compared with multiple tracking location variants of SWEI. However, the assumption of a linear model leads to an overestimation of the shear modulus in viscoelastic media. A new reconstruction technique denoted single tracking location viscosity estimation (STL-VE) is introduced to correct for this overestimation. This technique utilizes the same raw data generated in STL-SWEI imaging. Here, the STL-VE technique is developed by way of a maximum likelihood estimation for general viscoelastic materials. The method is then implemented for the particular case of the Kelvin-Voigt Model. Using simulation data, the STL-VE technique is demonstrated and the performance of the estimator is characterized. Finally, the STL-VE method is used to estimate the viscoelastic parameters of ex vivo bovine liver. We find good agreement between the STL-VE results and the simulation parameters as well as between the liver shear wave data and the modeled data fit.