I'm trying to count the number of distinct lines with square, non-square or zero norm in a 3-dimensional orthogonal space over $\mathbb{F}_q$ for $q = p^n$ an odd prime power. We know there's only one symmetric bilinear form in 3-dimensions up to equivalence, so I've assumed that the gram matrix of the form is the identity $I_3$.
I'm pretty sure there are $q\frac{q-1}{2}$ lines with square norms and $q \frac{q+1}{2}$ of non-square norms when $q \equiv 3 \mod 4$ and vice versa for $q \equiv 1 \mod 4$ with the remaining $q+1$ lines being isotropic, but I have no idea how to actually get any of these numbers.
If it's any easier to figure out, I'm actually trying to determine the number of 2-spaces which are of $+$ type, of $-$ type or degenerate within an orthogonal 3-space... I hoped counting lines was a better route.
Any help would be much appreciated!
It's much easier to consider working in the projective plane $\mathrm{PG}(2,q)$ (which has underlying vector space $\mathbb{F}_{q}^{3}$). In this case your orthogonal space is a conic of the projective plane, that is, your one-dimensional isotropic spaces correspond to a set of $q+1$ points in the plane, no two of which are collinear.
The number of nonisotropic points ($1$-spaces) that correspond to squares/nonsquares is, as you said, $\frac{q(q+1)}{2}$ and $\frac{q(q-1)}{2}$, though which of these numbers goes with the squares and which goes with the nonsquares depends on the quadratic form (yes they are all equivalent, but consider the forms $Q$ and $\alpha Q$ for $\alpha$ a nonsquare to see that this is not invariant). If you need to show this (I don't think this helps your problem relating the types of 2-spaces), it's a little bit complicated; you can consider a point as being "interior" to the conic if it lies on no tangent to the conic, and "exterior" if it lies on exactly two tangents. This partitions the points, and you can show that each set of points forms an orbit under the group of the conic (which preserves squares/nonsquares of the quadratic form).
For what you want to show about the 2-spaces (which are lines of $\mathrm{PG}(2,q)$), you just want to argue geometrically about the number of tangents ($q+1$), secants ($\frac{q(q-1)}{2}$), and exterior lines ($\frac{q(q+1)}{2}$) to the conic.