The full error indicates that some private API is used.
I used the env variable LD_DEBUG to tease it into telling us more:
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=python2 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/libpthread.so.0 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/libc.so.6 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/libpython2.7.so.1.0 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/lib64/ld-linux-x86-64.so.2 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/libdl.so.2 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/libutil.so.1 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/libm.so.6 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/calibre/calibre/plugins/libheadless.so [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/libfontconfig.so.1 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/libfreetype.so.6 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/libglib-2.0.so.0 [0]
15635: symbol=_ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb; lookup in file=/usr/lib/libQt5Gui.so.5 [0]
15635: /usr/lib/calibre/calibre/plugins/libheadless.so: error: relocation error: symbol _ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb version Qt_5_PRIVATE_API not defined in file libQt5Gui.so.5 with link time reference (fatal)
15635:
15635: file=/usr/lib/calibre/calibre/plugins/libheadless.so [0]; destroying link map
15635:
15635: file=/usr/lib/libfontconfig.so.1 [0]; destroying link map
Cannot load library /usr/lib/calibre/calibre/plugins/libheadless.so: (/usr/lib/calibre/calibre/plugins/libheadless.so: symbol _ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb version Qt_5_PRIVATE_API not defined in file libQt5Gui.so.5 with link time reference)
QLibraryPrivate::loadPlugin failed on "/usr/lib/calibre/calibre/plugins/libheadless.so" : "Cannot load library /usr/lib/calibre/calibre/plugins/libheadless.so: (/usr/lib/calibre/calibre/plugins/libheadless.so: symbol _ZN20QPlatformIntegration11screenAddedEP15QPlatformScreenb version Qt_5_PRIVATE_API not defined in file libQt5Gui.so.5 with link time reference)"
qt.qpa.plugin: Could not find the Qt platform plugin "headless" in "/usr/lib/calibre/calibre/plugins"
This application failed to start because no Qt platform plugin could be initialized. Reinstalling the application may fix this problem.
Available platform plugins are: headless (from /usr/lib/calibre/calibre/plugins), wayland-org.kde.kwin.qpa, eglfs, linuxfb, minimal, minimalegl, offscreen, vnc, wayland-egl, wayland, wayland-xcomposite-egl, wayland-xcomposite-glx, xcb.
[1] 15635 abort (core dumped) QT_DEBUG_PLUGINS=1 LD_DEBUG=all ebook-convert ward.html ward.mobi --authors
Your error includes reference to /usr/lib/calibre/, thus this is not the
prebuilt binaries. (Also, the prebuilt binaries do not use Qt 5.13 anyway.)
Is this a distribution package or self-compiled? Arch Linux has built
calibre using qt5.13 since June 19 and it works fine. However, calibre
does need to be rebuilt with qt5 changes, because it links to the
private headers. And your problem plugin is libheadless.so, which is one
of those private-ABI components. Have you tried rebuilding calibre to
see if that fixes things?