500 relics..a little bit of a grind, but not bad.
Every expedition has 3 parts or floors to the expedition for a total of 9 relics.
Each floor will track relics per part. 33/66/100%. that way you can easily keep track so see if you have them all.
Only use the portal to the next part once you have all 3 relics on that section.
Relics can be tracked in Journal > Achievements > Quests > Relic Hunter (I, II, II)
Finishing off the campaign expedition quests before you unlock the dailies will give you 45/500 relics.
Once you complete the Master Expedition quest you will be able to unlock the expedition dailies, which can be repeated 3 times a day. (27 relics a day)
This should take around 17 days of dailies (assuming you miss no relics)
Tested - you can do as many as you want each day (by helping other), but you can only get 27 relics a day from dailies.
Jappe confirmed you can not hold onto the quest, do more runs and get additional relics. (Thanks for testing this theory Jappa)
Updated note : If you are in a party, it doesn't matter who collects them as it will still count for you. (but only on your first 3 dailies, not after that)
Update note number two :
Seems like the maps with the bridge and lava, can sometimes bug out only giving you 8 for the run (3rd piece cant be found) - not always the case though. Seems like the map with the bridge there is one close to the camp fire at the start - but it blends in and give people issues - double check this one otherwise its a lot of running back