I guess I'm asking multiple questions here. Where did the tale of the Fountain originate? Why did the Spanish try to find something that was a myth? When did they stop searching for it? What kind of conflicts did the Fountain produce?
One of my early mentors and a good friend is Dr J. Michael Francis, currently a chaired professor of Florida Studies at the University of South Florida. He gives lectures on this sort of thing all the time. He is undoubtedly the top La Florida historian today. Here is one of his talks that he gives on the subject:
EDIT: The link is Dr Francis's presentation on this topic where he not only answers this question but also goes into the expansion of the myth into United States popular memory.
Did they really Have that firm belief? I'm from Spain and I haven't heard any mention to that legend.
The only close legend I've heard they were interested in was about el dorado.