If you're looking to create "one library and in the darkness bind them" ;) - you want to archive the dependent libraries into yours.
Unlike with shared libraries, a static library is a loose collection of object files - this means that when your library code referencing other libraries - whoever uses your library is expected to have those other libraries as well.
On *nix systems you can have accomplish this with a 'post link' (even though there's no actual linking) via something like:
QMAKE_POST_LINK += libtool -static -o $$DESTDIR/myaggregatedsdk.a $$DESTDIR/mysdk.a $$PWD/../dependencies/libs/libsomeotherlibrary.a $$[QT_INSTALL_LIBS]/libQt5Core.a
That made up example combines your sdk library, some other library, and just as an example of something a bit more advance - how to aggregate a Qt static library into your library (if you have build the static libraries.)
On Windows using Visual Studio's nmake system - it's slightly different - you don't do it after you build, instead you pass some flags to nmake and it does it during the build process:
QMAKE_LIBFLAGS += /OUT:$$TARGET "$$PWD/../dependencies/libs/somelibrary.lib" "$$[QT_INSTALL_LIBS]/Qt5Core.lib" "$$[QT_INSTALL_LIBS]/Qt5Network.lib"
Be careful doing this type of thing though because you'll likely need to have many versions of your library available (especially on Windows.)
There's a reason that this is an unusual request - and sometimes it's the right thing to do, but if you're not ABSOLUTELY certain you need to do this - then you should avoid it and go one of two ways.
Either stick with providing your static library and listing the other libraries as dependencies that others must obtain in order to build - or build a shared library (which is quite often the best solution.)
Sometimes, when building an SDK, you must support all forms though (which is why I had to do this myself once.)
Hope this helps.