Is this comboBox editable?
if it is, you can simply create a derrived class that overrides mousePressEvent and emit a signal when widget under mouse isn't QLineEdit. If something is clicked in comboBox and it's not QLineEdit it must be an arrow button.

This sollution works fine in my application.

Take a note that if comboBox isn't set to editable it have no QLineEdit so what im doing in this case is setting comboBox to editable, but with installed eventFilter that blocks keyboardEvents, so combobox looks & works like editable but edition is blocked.