Yes. Some fates spawn near or on top of questing locations. One such example is fates south of Camp Dragonhead when you are doing the gathering Ixal quest there. They make it really difficult (impossible) to use one of the trees if your gatherer is lower level, until the fate is finished.

There are quite a few fates that spawn close to main story quest locations, too. But they are not as frequent (and with the timer on them, it's not hard to know when to come to not be caught by another fate again), not to mention, it's actually very much possible to pass the mobs for the locations without fighting them. It just takes some basic skill in paying attention to the surroundings, instead of standing in the first spot from which you can interact with whatever is part of the quest...

If desperate, just ask someone through a Yell to "tank" the mobs for a few seconds...It's not like they need to be killed. Just grabbing their attention so that you can interact. As a summoner/scholar, that can easily be done with the summon, just have them take aggro. As anyone else...well, the Chocobo companion can leave for a little bit. If you aggro only one mob, max two, you should be able to interact with the quest easily enough.