This paper focuses on the issue of asynchronous filtering for Markov jump Takagi-Sugeno (T-S) fuzzy neural networks (MJTSFNNs). We firstly consider the general transition probabilities (GTPs) including both uncertain parts and unknown ones. Then, by introducing stochastic variables, the filtering system involves not only packet loss but also quantization. After that, through Lyapunov stability theory and stochastic process knowledge, we design an asynchronous fuzzy filter and obtain a sufficient criterion to ensure the MJTSFNNs stochastically mean square stable with some performance. Finally, an example verifies the validity of our results.