I have compiled data and identified best fit quadratic equations ($R^2>99$%) in the $XZ$ and $YZ$ spaces like so:
$z = 0.0000004x^2 - 0.0035x + 26.588$
...where $y = 358.242$
$z = 0.0002y^2 - 0.207y + 62.77$
...where $x = 3750$
My assumption is that these two equations para-metrically define curves that are slices out of a 3D surface in perpendicular planes. I would like to do some analysis of this 3D surface (in actuality there are multiple of these surfaces that will interact with each other, but I don't want to complicate the question).
The question is: how can I take these two equations and turn them into a single equation (in terms of $X$ and $Y$) that defines a 3D surface?
Note: I did a search, but I am a civil engineer and it has been a long time since I have had to think about this kind of thing, so I am having trouble seeing which existing questions are most relevant to my problem.
Your curves cannot be slices of a same 3D surface, with equation $z=f(x,y)$. If we set $x_0=3750$ and $y_0=358.242$, then the equations of these slices should be $z=f(x,y_0)$ and $z=f(x_0,y)$, but this requires that $f(x_0,y_0)$ should have the same value on both slices. In your case this doesn't happen, because substituting $x=x_0$ in the first equation we get $z\approx 19.1$ while substituting $y=y_0$ in the second one we get $z\approx 14.3$.
I think you should instead run your best fit on $z=f(x,y)$, taking into account both sets of data.