I would do this:

First detect the equation with additive characters, then reverse the Fourier transforms on the $g$'s, then break into subsums
$$ \sum_{\{a,b,c,d \in \mathbb{F_q} : a^2+ab=c^2+cd \}} \hat{f}(a)\cdot \overline{\hat{f}(c)}\cdot \hat{g}(b)\cdot \overline{\hat{g}(d)} $$
$$ = \frac{1}{q} \sum_{\{a,b,c,d , \lambda \in \mathbb{F}_q \}} \psi( \lambda( a^2+ab-c^2-cd)) \hat{f}(a)\cdot \overline{\hat{f}(c)}\cdot \hat{g}(b)\cdot \overline{\hat{g}(d)} $$

$$= \sum_{a,c,\lambda \in \mathbb{F}_q } \psi ( \lambda (a^2-c^2))\hat{f}(a)\cdot \overline{\hat{f}(c)}\cdot g(\lambda a)\cdot \overline{g(\lambda c)} $$

Now for any $u \in \mathbb F_q$ not equal to $0,1,$ or $-1$, the sum over the subset where $c = a u $ and $a\neq 0$ is possible to control: It is
$$\sum_{a,\lambda \in \mathbb{F}_q , a\neq 0 } \psi ( \lambda a^2 (1-u^2) )\hat{f}(a)\cdot \overline{\hat{f}(a u)}\cdot g(\lambda a)\cdot \overline{g(\lambda a u )} $$
$$\sum_{a,b\in \mathbb{F}_q , a\neq 0 } \psi ( a b (1-u^2) )\hat{f}(a)\cdot \overline{\hat{f}(a u)}\cdot g(b)\cdot \overline{g(b u )} $$
which by Plancherel and Cauchy-Schwarz is bounded by

$$ \sqrt{q \sum_{a \in \mathbb F_q,a\neq 0 } \left | \hat{f}(a)\cdot \overline{\hat{f}(a u)} \right|^2 \sum_{b \in \mathbb F_q} \left | g(b)\cdot \overline{g(b u )} \right|^2} $$
so the sum over all such $u$ is bounded by

$$\sqrt{q} \sum_{u \in \mathbb F_q, u\neq 1,0,-1} \sqrt{ \sum_{a \in \mathbb F_q,a\neq 0 } \left | \hat{f}(a)\cdot \overline{\hat{f}(a u)} \right|^2 \sum_{b \in \mathbb F_q} \left | g(b)\cdot \overline{g(b u )} \right|^2} $$

$$ \leq \sqrt{q}\sqrt{\left(\sum_{u \in \mathbb F_q, u\neq 1,0,-1} \sum_{a \in \mathbb F_q,a\neq 0 } \left | \hat{f}(a)\cdot \overline{\hat{f}(a u)} \right|^2 \right) \left( \sum_{u \in \mathbb F_q, u\neq 1,0,-1} \sum_{b \in \mathbb F_q} g(b)\cdot \overline{g(b u )}\right)}$$

$$ \leq \sqrt{q} ||f||_2^2 ||g||_2^2$$

The remaining terms are those where $c=a, c=-a, c=0, a=0$, or all $4$ of these conditions are satisfied. These terms will be lower-order for generic $f,g$ but can lead to the main term for special $f,g$, and in any case their size can be estimated in the same elementary way.