I tried duplicating my data source and creating a calculated field using the following formula:
WINDOW_MEDIAN(median(datediff('day',[HUM_RSR_SUBMISSION_DT],[Initial Approval Date])))
(Per the solution posted here: http://community.tableau.com/message/174570)
This almost gets me what I want. The problem is that the median is the same (127.5 days) for all date periods (see attached screenshot). I want the overall median to be recalculated for each quarter. I think I'm close, I just need some help tweaking the Window_Median formula.