This quest was more or less a Week 5 quest, however, many could not complete it because of issues. It's back for Week 6, where you have to deploy Alien Nanites anywhere other than Holly Hatchery.
This is one of those quests that is incredibly simple, if you are lucky, as Alien Nanites spawn in chests and are pretty rare to boot. They can also be found in ground loot but are equally as rare. There is a higher chance of finding them if looting in areas where Abductors spawn.
If you complete the mini-battle game after being abducted onto the Mothership inside the loot room, you also have a chance to find Alien Nanites.
Once you have acquired Alien Nanites, to complete the quest, all you have to do is throw them like a grenade. This will create a section of low-gravity environment for anyone who gets caught inside, which will be quite jarring if unexpected.
A great place to find them was Holly Hedges, but it seems for this Week 6 quest, you have to find them anywhere else.