Hey,
The only reasons Xur would not have the quest for you is if you've already completed the quest this week or your character already has the quest.
Since you haven't been able to pick up the quest for a while the second option is the most likely. You'll have to complete that quest before Xur will give you another.
If your character does not have Xenology in their inventory, then you've likely abandoned it. Unfortunately, the quest is not appearing in the Quest Archive, so the only way to complete it is to pick it up with another character.
If you are unable to pick it up with any of your characters, then your best option might be to wait for Bungie to change the quest so that it can be recovered once abandoned.