The romance got its sea legs in the books (and fan fiction)
It’s hard to pinpoint exactly when Jon/Dany became a popular fan romance. Martin’s first book in the series, A Game of Thrones, was published in 1996 and predates current internet hysteria. But, as with all true romance fandoms, you can trace origins through fan-fiction archives. This Livejournal fic, “The First,” dates back to 2004, between the release of the third and fourth books in Martin’s saga. The popular fan-fiction site, An Archive of Our Own, has Jon/Daenerys romance stories starting from around 2007, four years before the show began. It’s possible even more stories exist in the annals of extinct Angelfire URLs and long-dead message boards.
Even before fan fiction popularized the idea, the book text indicated a future romance. Dany has visions in the House of the Undying of blue roses, Lyanna Stark’s favorite flowers, which were given to her by Rhaegar as a loving gesture. Dany sees blue roses growing out of ice, perhaps indicating her own romantic union with a Stark. The blue rose has become a potent symbol in the Jon/Dany affair, popping up in fan art and even, briefly, on the show, when Daario presented her with several different flowers, specifically a blue rose.