Neutrinos play a fundamental role in the understanding of the origin of ultra-high-energy cosmic rays. They interact through charged and neutral currents in the atmosphere generating extensive air showers. However, their a very low rate of events potentially generated by neutrinos is a significant challenge for a detection technique and requires both sophisticated algorithms and high-resolution hardware. A trigger based on a artificial neural network was implemented into the Cyclone® V E FPGA 5CEFA9F31I7 - the heart of the prototype Front-End boards developed for tests of new algorithms in the Pierre Auger surface detectors. Showers for muon and tau neutrino initiating particles on various altitudes, angles and energies were simulated in CORSICA and OffLine platforms giving pattern of ADC traces in Auger water Cherenkov detectors. The 3-layer 12-8-1 neural network was taught in MATLAB by simulated ADC traces according the Levenberg-Marquardt algorithm. Results show that a probability of a ADC traces generation is very low due to a small neutrino cross-section. Nevertheless, ADC traces, if occur, for 1-10 EeV showers are relatively short and can be analyzed by 16-point input algorithm. We optimized the coefficients from MATLAB to get a maximal range of potentially registered events and for fixed-point FPGA processing to minimize calculation errors. New sophisticated triggers implemented in Cyclone® V E FPGAs with large amount of DSP blocks, embedded memory running with 120-160 MHz sampling may support a discovery of neutrino events in the Pierre Auger Observatory.