This paper deals with the relaxation stability analysis and event-triggered controller design of discrete-time positive Takagi-Sugeno (T-S) fuzzy networked control systems (FNCSs). The constraints of positive conditions in NCSs bring significant challenges to stability analysis and controller design. A 1-norm event-triggered scheme is proposed for the positive system, which can effectively reduce the frequency of network communication. Since the event-triggered T-S fuzzy controller (ETTSFC) can only observe the latest transmission state after the triggering action, the ETTSFC selects the premise variables different from the T-S fuzzy model, improving design and implementation flexibility. Using the linear copositive Lyapunov function (LCLF), the design criteria of the event-triggered controller to ensure the positivity and stability of the closed-loop system are derived. To relax the sum-of-squares (SOS) based conservative condition, we adopt the Taylor series membership functions (TSMFs) and introduce TSMFs and their approximation error into the stability conditions. Finally, an example is given to illustrate the effectiveness of the design.