I think there are many pieces from CotSQ that can be looted to expand a GDQ campaign -- most of part 2, for example, and there are plenty of maps from other places than can be used. I was going to use a fortress from CotSQ as the Eilservs estate, for example. The ELs in CotSQ are approximate to a straight-up conversion of the D series.
The biggest difficulty with running CotSQ as part of a GDQ sequence is that its basic plot is intended to be one big curve ball, turning what most of the players know about drow upside down; the D series presumes that the PCs know next to nothing about drow.
If I were determined to combine them, I would probably work it like this: forget about running Q1. Once the PCs reach Erelei-Cinlu and have taken care of business there, have them be approached by representatives of either the Temple of Lolth or House Eilservs (whatever comes the closest to being considered an "ally") with a proposition: A distant drow city to the west has gone silent, no merchant trains have come or gone. Maybe the PCs could investigate? And so on, and so forth. The employers, of course, will have suspicions as to what is happening, but they will withhold this information from the PCs.
