Temporal Acquisitions Specialist

You obtain quest starter items for this achieve from inside the Time Rifts (hourly Tyrhold Reservoir event added in 10.1.5)

All of the items were just randomly spawning either in p1 on some events or in p2 after the boss is defeated. There are a total of 8 items from p1 and 6 from p2 (2 per "Azeroth" total).

P1 starters are random drops - they show up on your map as quest items or npcs, that you can pick up or talk to. You don't need to be the one killing mobs, they will show up anyway. Some of them dropped on the ground from random trash, some from mini-bosses and some were from npcs that spawned. P1 contains mobs from 2 "Azeroths". i.e. it can be Mecha + Void

P2 starters show up after the boss is defeated. Like p1, some are items on ground and some are npcs. There can be a slight delay between boss dying and item spawning - but less then 1 min. Not sure if it's 100% drop chance, but my last 4 after hotfixes were all there. Wasn't the case before, so need some confirmation. P2 has boss from one of the "Azeroths" from p1. i.e. for my previous example - it will be either Mecha or Void for p2.

Each item drop chance is separate, i.e. you can get multiple on the same run (I got both 2 x p1 and p1 + p2 combinations)

Here are the items per "Azeroth":
Azmourne - Scourge:
Corrupted Ashbringer The Sins of the Ashbringer from miniboss in p1,
Gilded Vase All That Glitters... on top of Lich King in p2

A.Z.E.R.O.T.H. - Mecha:
Boffins Buzzing Boffins from p1 friendly spawned npc,
Mechanized Cog Cogs in a Machine from running miniboss in p1 [npc=206156] (you need to facecatch him to deal proper damage)

Azmerloth - Murlocs:
A Fish A Fishy Gift from p1 friendly Murloc You,
Azmerlothian Hourglass A Murloc and a Dragon Walk into a Time Rift... from murloc Alexstrasza on your way to portal in p2

Azewrath - Fel/Demons:
Map to Suramar The Resistance from p1 friendly npc that spawned after treant miniboss died,
Antlered Skull An Antlered Skull from Illidan ship's furthest corner in p2

Azq'roth - Void/Black Empire:
Amber Blade Mantid Methodology from random Klaxxi drop in p1,
Chipped Stone Tablet A Stony Reception at the back of the room in p2

Warlands - Alliance/Horde:
Blood Soaked Book Wasteland Warriors from random npc in p1,
Oil-Soaked Log Book The Endless Conflict on airship in p2 (in captains quarter)

Ulderoth - Titans:
Titanic Disc Titanic Beasts random p1 drop,
Watcher's Log Dangerous Documentation next to one of circular areas in p2 (~ 58, 39)

The toy you get in the end summons a little flying robot head following you for 5 min

Here is the time rift rotation

Azq'roth - Old God
Warlands - Alliance/Horde
A.Z.E.R.O.T.H. - Robot
Azmerloth - Murloc
Ulderoth - Titan
Azmourne - Scourge
Azewrath - Demon

There's always two types of invader mobs each rift but the second mob type seem to be random so hover over the horn icon on your map 5 minutes before it starts to see which mobs are going to be up in case you need any of them for the achievement.

Example:
If horn icon says
-Murloc Invaders
-Demon Invaders
The first one always determines which rift it will be so in this case it would be Azmerloth.

Check out Wuod's comment for more details and how to get the quests for the achievement.

Worth noting that after turning the items in to Soridormi, you can find them on the tables behind her (near [npc=207715]). Clicking on each will reveal a bit of lore about the item and/or the alternate timeline it came from.