No, there isn't a limited time window to do his quest. It just might be hard to trigger certain conditions for it to progress.
The game isn't designed to accommodate not having either, as that's actually more work than you might expect. Right now the game understands that you have at least cock or a pussy, so only needs to accommodate for two situations. Not having either means a third situation needs to be accounted for in basically all the genitalia focused scenes in the game.