Unseen Influence

A good way to get more than half needed quickly is to go to Fang'rila, and kill the three champions. You get a few for each kill, but only once every day. You can stick around the little arena and just kill whatever other people summon, since they are free tap.
To be more precise, the minor champion gives 1x, the major 2x and the prime 3x, for a total of 6x Fragments.

To complete this quest, i would suggest you do not focus on it and do the other dailies. While doing them, you will almost inevitably stumble upon a few rares. It happened to me a few times to not even have to bother doing the champions!

Finally, you can get the item [item=128502] to help you find rares.

This is a horridly designed quest that might have seemed fine on PTR with a limited playerbase but on live with the entire server running around doing their dailies it just scales horridly. Right now its fine since there are plenty of 1 time find treasures to make up the difference but once you've exhausted those you are stuck with hoping someone hasn't just rolled through the bonus area you pick killing all the rares and picking up the respawnable chests. Very very poorly thought out quest.

This is another of those absurdly annoying quests that takes forever, but there are some tricks to get around it.

Normal questing for these can honestly take hours, because you'll only find them on rares, treasures, and a couple other places.

You can get one out of most of the treasures you'll find, but after the treasures are gone, that approach won't work.

The fastest way to get to your first (6) [item=128346] is to go to the Blackfang Challenge Arena at (54,81), and kill the three champions as they are summoned one by one. On populated servers, you can just wait for others to purchase the items that summon each champion and down them, this past week you'd see someone downing all three within about 15 minutes.

If you are on a less populated server, you can purchase the Challenge Totems listed below from Z'tenga the Walker at (55,74), the entrance to the Fang'rila area. WARNING: If you are not geared up, do NOT enter Fang'rila without friends, the NPCs there are very aggressive and hard to kill!. If you are geared up, or you have a group, you'll need to farm the Blackfang NPC's in the area, they drop the Blackfang Claws used as currency with Z'tenga to buy the totems. Activate the totem and summon Champ inside the Challenge Arena by clicking totem, down Champ, take Fragments. Note: for whatever reason, it takes a minute or so before you can do your next totem, so just be patient.

[npc=92766] = [item=124093] = 5 [item=124099] = 1 [item=128346]
[npc=92817] = [item=124094] = 10 [item=124099] = 2 [item=128346]
[npc=92819] = [item=124095] = 25 [item=124099] = 3 [item=128346]

There are your first six Fragments. Don't try to buy more totems to resummon the same Champion for more Fragments, you can only loot them once per day. Same thing goes for waiting for others to summon, if you've already killed one of the Champions, that's it till tomorrow.

Then for your 7th and 8th guaranteed Fragments go get [object=241712]. This chest respawns, and each day you can loot another fragment, one of a few treasures that does this. Go to cave entrance at (37,76), and head inside. You'll have to climb up on the platform at (35,78), it's a little tricky, but you'll figure it out. Open that chest up, and [npc=93125] will pop out. He's not overly difficult. Kill him, and loot the Fragments. Basically he's a guaranteed rare that will always be there, respawn time is like 2 minutes. Like the Champions above, don't try to kill him again for another Fragment, you only get one per day. (I've heard some people only get one from this, I've gotten two every time.)

A huge must to have in this area regardless are [item=119093] and [item=128502]. With the feather you can fly around up high and search out rares, if you find one, land close as you can by circling down, then activate the crystal to teleport directly to the rare, and kill it. Sometimes it teleports you to a different rare, which, if you are hunting a specific rare, that stinks, but if you are just after these crystals, no biggie, kill rare, get crystal, and then you might even be able to get to the other rare you saw before someone kills it.

EDIT: HUGE help to get the last two: Once you reach revered with the Order of the Awakened, you can purchase the [item=128503]. Now be warned, seems like half the time it sends you over to Varyx the Damned (requires several people to break the beams rare), but the rest of the time you can get your last two, 15 minute cooldown on the item. Here's another tip: You spend 4,250 gold to get this item, along with trading in your [item=128502]. Notice though, there is a two hour window and you can sell that item back to the vendor. So use it each day to get the crystals, then sell it back. There really isn't much reason to have it other than going after rares, once you have the Jungle Stalker and Exalted with Order of the Awakened, you aren't likely going to need the [item=128503] any longer, plus you'll still have your [item=128502] anyway for finding rares nearby.