In order to solve the time-varying quadratic programming (TVQP) problem more effectively, a new self-adaptive zeroing neural network (ZNN) is designed and analyzed in this article by using the Takagi-Sugeno fuzzy logic system (TSFLS) and thus called the Takagi-Sugeno (T-S) fuzzy ZNN (TSFZNN). Specifically, a multiple-input-single-output TSFLS is designed to generate a self-adaptive convergence factor to construct the TSFZNN model. In order to obtain finite- or predefined-time convergence, four novel activation functions (AFs) [namely, power-bi-sign AF (PBSAF), tanh-bi-sign AF (TBSAF), exp-bi-sign AF (EBSAF), and sinh-bi-sign AF (SBSAF)] are developed and applied in the TSFZNN model for solving the TVQP problem. Both theoretical proofs and experimental simulations show that the TSFZNN model using PBSAF or TBSAF has the property of converging in a finite time, and the TSFZNN model using EBSAF or SBSAF has the property of converging in a predefined time, which have superior convergence performance compared to the traditional ZNN model.