Generalised coordinates

The $q_{nm}$ are the amplitudes of the normal modes of vibration of the plate with sides $2\pi a$, $2\pi b$. He is using the usual plate wave equation $$ \frac{\partial^2 y}{\partial t^2}= D \left(\frac{\partial^2 y}{\partial t^x}+\frac{\partial^2 y}{\partial z^2}\right)^2 $$ and expanding the displacment as $$ y(x,y,t)= \sum_{n,m} q_{n,m}(t) \cos (mx/a)\cos (ny/b) $$

I think q here denotes a position coordinate. There exists a particular form of potential in which the dynamics of the ball work. It might be that, $q_{mn}$ determines the position of the $m^{\text{th}}$ particle of the ball w.r.t to the $n^{\text{th}}$ particle of the plate. At first, the total potential energy of the $m^{\text{th}}$ particle of the ball due to its interaction with each and every particle of the plate is calculated; that's how the $\sum_{n}$ comes. Next the potential energy of the entire ball is calculated by adding the potential energies of each such particle of the ball; this is how the $\sum_{m}$ comes. However, as both the ball and the plate have continuous mass distribution, the summations should be replaced by integrals.

${\dot q}_{mn}$ defines the velocity of the $m^{\text{th}}$ particle of the ball w.r.t the $n^{\text{th}}$ particle of the plate. While calculating the kinetic energy, the summations appear following a similar argument as in the above paragraph and as it seems, they also should be replaced by integrals.

I do not know the exact context of your question though. This is as far as I got from it.