Connect the drive and note its volume name.
Open a terminal and type the following:
find "/Volumes/volumeName/" -type f | wc -l
The first command lists all entries of type "file" on the volume, one per line, and then sends the result to the second command which counts the number of resulting lines. Note that the result will be misrepresenting if you lack access to some directories on the drive or if for some strange reason any file names contain line breaks.