For example, if I have two numbers
$m=\text{max power of each coefficient}$
$n=\text{max sum of the power of the coefficients}$
so for $~m=2~$ and $~n=3~$, the polynomial(which consists of two variables $~x~$ and $~y~$) would look like
$=c_{1}+c_{2}x+c_{3}y+c_{4}x^2+c_{5}y^2 +c_{6}xy+c_{7}xy^2+c_{8}x^2y$
Is there a good way to find the number of terms in the polynomial.
Assuming my interpretation above is correct, after rephrasing and generalizing further the number of available variables (and renaming the variables from $x$ and $y$ to instead $x_1,x_2,\dots,x_k$) you are trying to count the number of non-negative integer solutions to the system:
$$\begin{cases}x_1+x_2+\dots+x_k\leq n\\ 0\leq x_i\leq m~~\forall i\end{cases}$$
If so, then introduce an additional variable, I'll call $y$ equal to $n-x_1-x_2-\dots-x_k$ and you now are working with the system $$\begin{cases} x_1+x_2+\dots+x_k+y \color{red}{=}n\\0\leq x_i\leq m~~\forall i\\ 0\leq y\end{cases}$$
From there, apply inclusion-exclusion over the events that an $x_i$ exceeded the value of $m$ and use stars-and-bars to calculate each case.
You get as a result a total of:
$$\sum\limits_{i=0}^k (-1)^i\binom{k}{i}\binom{n-i(m+1)+k}{k}$$