Actually, slight correction. Turning the relics in doesn't immediately start the quest. You just unlock the questline for it but don't actually obtain it. You actually have to wait for The First Firewall mission to become available for one day only(which is the 7th of each month). Doing that classified mission is how the quest actually begins. Only way to guess if you have them all on that character is if you don't get anymore.

For me personally they all dropped on patrol while doing bounties. No need to check how many you have, since the mission will start after handing the fourth one in . Knowing how many you already have, wont help the next one to drop any quicker. Just keep doing patrol etc and you will eventually get them all. Obviously the more you play the sooner you will get them ;)

I'm not sure as I didn't pay attention to how they dropped, but I slightly remember a Quest where when you turn in one to the Gunsmith, he will ask you to grab 5 more and turn them in. Once you do that, just wait until the mission is available.