Hi again Hernandez I see that you have changed your question a little. Now that you have only 'quest' (and have removed 'and actions'), it seems that you are implying there is only one thing (the quest) that belongs jointly to both the groups, so therefore you should only use one apostrophe. Personally, I think that apostrophes should be banned. I mean they may be okay in principle, but, given that most of the world (including native speakers) dont know how to use them properly, theyre effectively rendered obsolete , and its about time that the grammar pilgrims and puritans give up their quest to force everyone to learn about them! (I am joking...sort of)

Hello Hernandez! If the religious quest of the puritans and pilgrims is the same, then you will use one apostrophe. If the religious quests differ, then there will be one apostrophe per group.

please