A variant of Abhinav Kumar's solution again uses his reduction to looking at \begin{equation}\tag{1} (9g^4+6g^2+1)h^4-8gh^3+(6g^4+12g^2+2)h^2+(-8g^3-8g)h+g^4+2g^2+1. \end{equation} Note that this is the product of the complex conjugate factors \begin{equation}\tag{2} 3g^2h^2 - g^2 + 4gh - h^2 - 1 - 2\sqrt{-3}gh(g-h) \end{equation} and \begin{equation}\tag{3} 3g^2h^2 - g^2 + 4gh - h^2 - 1 + 2\sqrt{-3}gh(g-h) \end{equation} So this shows non-negativity of (1). Furthermore, if (1) is zero, then both factors (2) and (3) vanish. Suppose that (2) has a rational solution. Then $gh(g-h)=0$ by irrationality of $\sqrt{-3}$. But the cases $g=0$, $h=0$, and $g=h$ yield $h^2+1=0$, $g^2+1=0$, and $(g^2+1)(2g^2-1)=0$, respectively. None of these has a rational solution.

A variant of Abhinav Kumar's solution again uses his reduction to looking at \begin{equation} (9g^4+6g^2+1)h^4-8gh^3+(6g^4+12g^2+2)h^2+(-8g^3-8g)h+g^4+2g^2+1, \end{equation} which equals \begin{equation}\tag{1} \left(3g^2h^2 - g^2 + 4gh - h^2 - 1\right)^2 + 12\left(gh(g-h)\right)^2. \end{equation} So this shows non-negativity. Furthermore, if (1) vanishes, then $gh(g-h)=0$. But the cases $g=0$, $h=0$, and $g=h$ yield $h^2+1=0$, $g^2+1=0$, and $(g^2+1)(2g^2-1)=0$, respectively. None of these has a rational solution.