I like the spinoff, but it would be the other way around.
Jellico would never end up doing exactly what Q wanted and Q would be always frustrated. Every episode would end with Q destroying Jellico in some spectacular fashion while Jellico smirks condescendingly with a quip “Go ahead and destroy me Q. Prove to both of us how weak you are that a simply human got the best of you”. Each subsequent episode would completely ignore the continuity and Jellico would be there like he was never destroyed. After 6 or 7 episodes even Q would notice this and start question things. Season (Series?) finale would reveal that all the other Q keep resurrecting Jellico because it keeps Q out of their hair.