This will depend on the photonic crystal: if the unit cell is rectangle, or other geometry with much less change in shape like a circle, you may try EME. Otherwise, FDTD might be more efficiet, in particular when the source is broadband.
Quality factor can be measured with relatively short simulation time in FDTD, as shown in online examples. EME is a frequency-domain method. If you want broadband and calculate Q from transmission, you can, by sweeping the wavelength using the built-in wavelength sweep, provided the mode properties do not change significantly over the spectrum. However whether it can actually find correct Q over million will depend on several factors.