Yes, they did. While early expeditions (like Ponce de León in 1513) were exploratory, the Spanish founded St. Augustine in 1565—the oldest continuously inhabited European settlement in the continental United States. Earlier attempts (e.g., Pensacola 1559) failed due to hurricanes and supply issues, but St. Augustine succeeded and became the capital of Spanish Florida.
So, contrary to the premise, Spanish colonization of Florida began in earnest in the mid-1500s, not the early 1500s, with St. Augustine as the enduring foothold.