Consider the two-dimensional wave equation:
$$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\left[\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right]$$
for some $c>0$. I have used separation of variables to solve for $u(x,\,y,\,t)$ in the rectangular region $0<x<a$ and $0<y<b$ subject to $u=0$ on the boundary of the region to get the solution
$$u(x,\,y,\,t) = \sum^{\infty}_{m=1}\sum^{\infty}_{n=1} \left(A_{m,n}\cos{\omega_{m,n}t}+B_{m,n}\sin{\omega_{m,n}t}\right)\sin{(q_{n}x)}\sin{(q_{m}y)}$$
Where $\omega_{m,n} = c\sqrt{q_{n}^{2}+q_{m}^{2}}$ and $q_{n}=\frac{n\pi}{a}$, $q_{m}=\frac{m\pi}{b}$.
I am asked to find the fundamental (lowest frequency) mode for this problem and to give its frequency but I'm not actually sure what this is. Can I have any hints?
The (angular) frequency of a separated solution is the coefficient of $t$ inside the trigonometric functions involved in it. That is $\omega_{m,n}$ in your notation. The formula for $\omega_{m,n}$ tells you that it's smallest when $m=n=1$.
By the way, the proper way to write the series you have is $$u(x,\,y,\,t) = \sum^{\infty}_{m=1}\sum^{\infty}_{n=1} \left(A_{m,n}\cos{\omega_{m,n}t}+B_{m,n}\sin{\omega_{m,n}t}\right)\sin{(p_{n}x)}\sin{(q_{m}y)}$$ that is using $p_n$ (or another letter) instead of $q_n$. The reason is that $p_n=\pi n/a $ and $q_m=\pi m/b$ are not the same thing: e.g., $p_1\ne q_1$ in general.
Summary: the fundamental mode is $\sin(p_1 x)\sin(q_1 x)$ and its frequency is $\omega_{1,1}$.