Is there any difference between the convolution operation applied to images and applied to other numerical 2D data? Is there any difference between the convolution operation applied to images and applied to other numerical 2D data?

For example, we have a pretty good CNN model trained on a number of $64 \times 64$ images to detect two classes. On the other hand, we have a number of $64 \times 64$ numerical 2D matrices (which are not considered images), which also have two classes. Can we use the same CNN model to classify the numerical dataset?
Short answer is no. You can't use a model trained for one task to predict on a totally different task. Even if the second task was another image classification task, the CNN would have to be fine tuned for the new data to work.

A couple of things to note...

1) CNNs are good for images due to their nature. It isn't necessary that they'd be good for any 2-dimensional input.  

2) By 2D numerical data I'm assuming you don't mean tabular data.