I mean the official installer aka apt/apt-get/dpkg.
These said 1236 packages are compiled in one block by the Ubuntu CI/CD pipeline. The ordering here is crucial as some packages depend on others. So, you get these packages built step by step.
If you side-load another version of Qt the other ~1200 packages are still
- using the officially build Qt version
- have a package dependency on said Qt version.
Thus, in your system, there will be 2 Qt versions present most probably. I am not telling you that this is your problem but warn you that you have to keep that in mind, also when starting the built binary.
Maybe you look into the files in /home/markus/Qt/5.15.2/gcc_64/lib/cmake. This file should contain the required compiler and linker flags to build against the library and use it accordingly. Have a look for the string Qt/5.15.2/gcc_64/include/QtGui and Qt/5.15.2/gcc_64/include/QtCore with some -I prefix. I suspect it will contain -I/home/markus/Qt/5.15.2/... but could also be -I~/Qt/5.15.2/... or similar.
You could also try to add the -I parameter manually (to make the compiler find the included files manually). In this case my fist point gets more interesting: Linking requires also some flags eventually.