I needed to compile gpsbabel myself because Ubuntu 16.04 only contains version 1.5.2, and GeoJSON support is only available in later versions.
It needs at least Qt 5.2, and I still had the 4.x development files installed (libqt4-dev). I removed them and installed the Qt5 development files (qtbase5-dev).
After downloading and extracting gpsbabel 1.5.4, I got an error:
$ ./configure ... qmake: could not exec '/usr/lib/x86_64-linux-gnu/qt4/bin/qmake': No such file or directory ...
The problem was that /usr/bin/qmake links to qtchooser, which still wanted to use Qt4.
After installing the qt5-default package, configuring gpsbabel worked and I could successfully make it.