This is normal. That is how Unix works. You can only execute binaries with their names in the PATH. You can double check with which
if they are found, or even locate
.
If you use the full path, then it is all clear for the interpreter for sure. I would suggest to do something in your bash, zsh or etc. profile:
PATH=~/install/Qt/bin/:$PATH
Alternatively, you can just do it manually in a separate session and work with Qt in there.
You could also create an alias for the full path if you wish, but that would be a stranger solution.
Of course, the best would be to install it system-wide, and once your distribution gets it packaged, just use your package manager.