tikz arrow package define where to put the base (starting point) of an arrow

Add anchor=west to the node options. You need the west anchor, not the south, because the node is rotated, and the anchors of the node are rotated along with it.

If your intention is to draw a fat arrow between the two functions though, you're better off using something like

  draw [line width=5mm,
         -{Stealth[width=15mm, length=10mm, inset=1mm]},
         blue]
         (-0.5, {G(-0.5)}) -- (-0.5, {F(-0.5)});

I think, where F and G are functions defined with declare function. (You don't have to do that though, you could write the function expressions directly in the coordinates, replacing x with -0.5.) The Stealth arrow tip is from the arrows.meta library, so that has to be loaded.

documentclass{article}
usepackage{pgfplots} % loads tikz, you don't have to do it explicitly
usetikzlibrary{shapes.arrows, arrows.meta}
pgfplotsset{compat=newest}
usepgfplotslibrary{fillbetween}

begin{document}

begin{figure}
  centering
  begin{tikzpicture}[
     every node/.style={single arrow, draw=none},
     declare function={
       F(x) = 4- sin(180*x);
       G(x) = 1+x*x;
       }
     ]
    begin{axis}[
        xmin=-1.0,xmax=1.0,
        ymin=-0.2,ymax=6.7,
        width=10cm,
        samples=100,
        domain=-1:1
      ]
      addplot[name path=A, thick] {F(x)};
      addplot[name path=B,thick] {G(x)};
      addplot[fill=gray,opacity=0.4] fill between [of=A and B];
      draw[dotted] (-1,0)--(-1,2);
      draw[dotted] (1,0)--(1,2);
      node [fill=green, single arrow head indent=1ex,minimum height=5cm,
      minimum width=2cm,rotate=90, anchor=west] at (-0.5,1.25) {};
      
      draw [line width=5mm,
             -{Stealth[width=15mm, length=10mm, inset=1mm]},
             blue]
             (-0.5, {G(-0.5)}) -- (-0.5, {F(-0.5)});
      
    end{axis}
  end{tikzpicture}
  caption{I need the arrwo to start in the bottom curve}
  label{subfig1}
end{figure}

end{document}