First fundamental form Wolfram MathWorld defines a paraboloid
and its differential parameters as
\begin{align*}
 P&=\left(\frac{\partial x}{du}\right)^2+\left(\frac{\partial y}{du}\right)^2+\left(\frac{\partial z}{du}\right)^2= \\
&=1+\frac{1}{4u} \\
 Q&=\frac{\partial x}{du}\frac{\partial x}{dv}+\frac{\partial y}{du}\frac{\partial y}{dv}+\frac{\partial z}{du}\frac{\partial z}{dv}= \\
&=\frac{1}{2\sqrt{u}}(\cos v - \sin v) \\
 R&=\left(\frac{\partial x}{dv}\right)^2+\left(\frac{\partial y}{dv}\right)^2+\left(\frac{\partial  z}{dv}\right)^2= \\
&=u \\
\end{align*}
Now, if these parameters correspond to the coefficients $E$, $F$ and $G$ described here, I don't understand how they arrived at the expression for $Q$.
 A: Despite other comments/answers, these quantities are the usual first fundamental form. Note that the Wiki link defines $g_{ij} = X_i\cdot X_j$. These are the usual $E,F,G$, and they are the dot products of the derivatives of the parametrization with respect to the independent variables. In your case the first parameter is $u$ and the second parameter is $v$, and we do in fact have
\begin{align*}
P&=X_u\cdot X_u=E,\\
Q&=X_u\cdot X_v=F, \quad\text{and} \\ 
R&=X_v\cdot X_v=G.
\end{align*}
I'm not sure why Wolfram is using different letters.
If you want a further reference, check out my differential geometry text.
A: The first fundamental form is the inner product of the tangent space in some point of the surface when you consider the surface contained in the ambient space $\mathbb{R}^3$. If you have a paraboloid $z=b(x^2+y^2)$, then the tangent vectors of the surface which generates the tangent space are
$v=[1,0, 2bx]$
and
$w=[0,1,2by]$
At this point the coefficients of the first fundamental form can be computed as follows
$E=\langle v, v \rangle=1+4b^2x^2$
$F=4b^2xy $
$G=1+4b^2y^2$
In your link about paraboloid, I guess the argument is a geodesic on the paraboloid.
