This depends on what you mean by "quality." If you are talking simply about the quality of the images that can be viewed on the monitor, then I would say that a cathode ray tube has higher quality than a liquid crystal display.
As you can see from the link that I have attached, CRT monitors are able to show much better images than LCDs. They are, for example, able to show colors in a way that is much more faithful to reality. This is a major aspect of quality.
If you have a broader definition of quality, however, LCDs may be more competitive. If quality includes portability and weight and such, then LCDs have a huge advantage. They are also less likely to flicker.
So there are many benefits and drawbacks to each -- follow the link for much more detail.