As far as know (it is possible that different concepts may be designated by the same name), Internal distance in proteins means the distance between two atoms that are part of a protein. It is closely related to the spatial disposition of the atoms in a protein.
The "internal distance matrix" is the matrix whose elements are (usually) the Euclidean distance between each pair of atoms that make up the protein chain.
I agree with Daniel, but still more interesting is the protein 'characteristic length' that corresponds to the average shortest path (ASP) that is the average length of the minimum path connecting any pair of distinct residues. The path length is expressed in 'number of contacts' and not in terms of euclidean distance, the basis is considering a protein as a network : the characteristic length is then an (inverse) measure of the efficiency of signal transduction inside the molecule see:
http://pubs.acs.org/doi/abs/10.1021/ci400218v
http://pubs.acs.org/doi/abs/10.1021/cr3002356
http://www.sciencedirect.com/science/article/pii/S0301462212000464
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