My Lexmark printer has these two print resolutions. Which is better?
1200 dpi
4800 CQ
It does not appear to be a dpi measurement, so the speculation in comments is probably not correct.
Lexmark CQ is a lexmark-specific "enhancement". I cannot quickly find what exactly CQ refers to but it is not a dpi-based measurement. The only source I can find that even attempts to describe it is from a press release circa 2005 which says "automatically adjusts dot size and shape simultaneously." This sounds a little like a hybrid between halftone and stochastic screening.
Very simplistically: stochastic screening looks a bit like dithering and creates tones by using a fixed dot size and varying the spacing of the dots. "Normal" (halftone) screening varies the size of the dots while maintaining a grid-like spacing.
One thing that seems to be the consensus about CQ is that the number used in Lexmark's settings is not in any way related to the max resolution of the printer head. Some of them 600x600 dpi or 1200x600 resolution and still have a 4800 CQ setting.
I do not know the brand or specifications, but it sounds like 1200 is for black only and 4800 is the array of color dots.