I would also suspect that any related user information to FileZilla would have been stored in your user profile directory/the ProgramData directory for FileZilla.
If you know the file name you're looking for, what I'd recommend is mounting your backup image set as a virtual drive as described above. Then, do a search for that file name and, depending on how you search/what you use, that mounted drive may be included in the search criteria for that file name.
BTW, off topic, but it's nice to see someone else program in FORTRAN!
While I haven't touched it since college in 1995, I knew that even back then it was an odd course to teach in a computer science major. It was more of an older language for business even back then, or so I thought. For my class project, I thought outside the box for FORTRAN and programmed a game: my version of Battleship.
EDIT: If you're feeling really adventurous/comfortable doing it, you could try this: take a current image of the partition where FileZilla is stored, which I'm guessing is the C: Windows one. Restore this image you're trying to restore this FileZilla file from. Do whatever you need to do with this FileZilla information while it's restored down, being sure to save this new file/information to somewhere other than the partition you just restored because you're about to restore the current image you took before the restore. Then, restore back the current image you took before restoring the FileZilla partition so your system is back to being live.
This is what I'd do, most likely, but I'm comfortable with juggling partitions. I've been doing it since about 1995.