You might just as well ask whether Microsoft gets any rights to your work for you having typed it into their word-processor, or whether Nikon gets rights to your book because a Nikon camera was used to take the photo that was later manipulated using the The Gimp into a cover that was added using Sigil, all on the an Ubuntu operating system (and I've still left many pieces of software used along the way). Same story going back a few years to ask if IBM, Olivetti or Corona got rights to books typed on their typewriters. Or ... well, you get the idea.
It's an interesting question, but the answer is a pretty clear: Calibre does
not get any rights in the work itself. (Note: Like most modern software, Calibre uses many other software components - see
here - and so the distribution of rights, if there were any, would be very very messy.)
Life might get more interesting with AI software evolving to a point where it might be claimed to be producing
art in its own right.