  The functional equations $ f^2+g^2=1 $ and $ f^2+2\alpha fg+g^2=1 $ are
respectively called Fermat-type binomial and trinomial equations. It is of
interest to know about the existence and form of the solutions of general
quadratic functional equations. Utilizing Nevanlinna's theory for several
complex variables, in this paper, we study the existence and form of the
solutions to the general quadratic partial differential or partial
differential-difference equations of the form $ af^2+2\alpha fg+b g^2+2\beta
f+2\gamma g+C=0 $ in $ \mathbb{C}^2 $. Consequently, we obtain certain
corollaries of the main results of this paper concerning binomial equations
which generalize many results in [\textit{Rocky Mountain J. Math.}
\textbf{51}(6) (2021), 2217-2235] in the sense of arbitrary coefficients.
