What the function of zener diode on this picture
They are protection of the gate. Yes it does not matter if Vgs goes below zero, but it matters if Vgs goes beyond the maximum rating of the mosfet, either in positive or negative.

That is why there are 2 zener diode connected in opposing direction.



EDIT

I implied that this circuit may have a positive or negative drive for the gate, which is not clear on the question. If there is supposed to be only positive drive on the gate, only a single Zener would be necessary as protection.