Normally, you never see the content of this register. But it leaks the info through in case of the SCF/CCF instructions. [https://worldofspectrum.org/forums/discussion/41704 Source]
Emulating Q is not strictly necessary. Only a A 1-bit flag, indicating whether the previous instruction computed flags, is needed enough to emulate the behaviour of SCF/CCF.
Also note that while POP AF and EX AF,AF' modify F, they do not compute new flag values. [https://github.com/hoglet67/Z80Decoder/wiki/Undocumented-Flags#scfccf Source]