Thank you again for your answers. Unfortunately I do not understand your last post. Could you please be more specific? From $f'_{m,n} = f_{m,n} + \frac{1}{1-a-b}$ and $f'_{m,n} = af_{m-1,n} + bf_{m,n-1}$ it follows $c = - \frac{1}{1-a-b}$, which is not necessarily right. What am I missing?

You are right, thank you! The first term is the one I am trying to simplify right now.

I see! You used the two examples $f_{m,0} = 2^m$ and $f_{0,n} = n+1$ from above. I added the general g.f. I found back then to the question. Looks pretty similar :)

Thank you for your answer. How did you get that generating function? It looks different to the one I have. Did you use some software? And shouldn't it depend not only on $a$, $b$ and $c$ but also on $f_{m,0}$ and $f_{0,n}$?

Perhaps "closed-form solution" is not the right term here. I know from one-dimensional linear recurrence relations (like $f_n = a f_{n-1} + b f_{n-2}$) that their solutions are very simple. I was hoping to be able to make it nicer. Maybe getting rid of the sum signs.