First of all I will like to know if there is a record of such instance. Some questions may arise from within a previous question or most often bible quotations and little comments to that question. Perhaps to give credit to the one who asked or demand for in-depth answers to that tiny part that strikes me, to what standard should my question with "inspired by the question..." be to avoid closure. The "inspiring" (previous) question may have very little or no answers that fully shed light on where my interest lies. Would it be seen as a duplicate?
All questions should stand alone on their own merit. It should not be necessary to go back and read through some previous question, answer or comment thread in order to make out what the question is about.
That being said, however the idea sparked in your head, there is no prohibition on forming questions out of whatever material you come across. You just need to do the work of forming it into a complete stand along question that can be answered in it's own right. Being tangentially related to a previous question on this site does not necessarily make it a duplicate.