First, make sure that the quest you're trying to do is allowed - check the editor text or pdf file (Appendix 3) for the Gods, Heros, and Quests. Each god has their own quest - if the quest you're trying to do is one that's listed under an "opponent" (unfriendly) god, it won't let you do it. Make sure you've got the Hero's Hall listed in white, not yellow in the "buildings allowed" section for that city. Make sure also that the Hero needed for that particular quest is allowed.
In the Mythology window of the editor, the opponent gods are in the left column, the proponent (friendly) gods are in the middle column, and the Sanctuaries allowed are in the right column. The first two gods (or maybe it's the first four) in the middle (proponent) column are the proximate gods - they're so close to your city that they will appear in it. I don't believe that you can have a quest set for any of them... it's the distant proponent gods that can have quests.
Make sure you have the proper Hero listed for the quest - according to the help file, *any* hero can fill any quest... there's no specific hero for each quest. So if you want Achilles to fill the quest, make sure that Achilles is one of the heros listed at the bottom of the mythology window.
Check your time settings - see if you've got the hero fighting a monster at the same time he's supposed to be filling a quest (it shouldn't really be a problem, but check anyway).
Make sure you have the quest set as a one-time only event.
Okay... the main things to check are: is the god issuing the quest a proponent (friendly) god?
Do you have the hero set up in both the mythology and quest event screens? Is it the same hero? If you want Achilles to do the quest, and Achilles is listed as Hero3 in the mythology window, make sure that Hero3 is the one set in the quest event window.
Make sure you can fill the requirements for that hero - if you want to use Achilles, but you have Armor shut off in either the city or world properties, then it won't work.
If the completion of the quest is supposed to trigger another event, make sure that the event number that you want to happen is listed after the 'Next Event' space, and that the event that you want to happen (after completing the quest) is a 'Triggered Only' event.
If none of that helps, then I'm durned if I know what to tell you... Wolfie.