We all know from "The Phantom Menace" that, about to die, Qui-Gon Jinn asked Obi-Wan to promise him to train Anakin because he believed the boy was the chosen one; the one who would bring balance to the Force. We also know that Qui-Gon Jinn told Mace Windu and Yoda that Anakin's cells had the highest concentration of midi-chlorians that he had seen in a life form. Was this previous reasonable enough ground to make Master Qui-Gon Jinn to declare that Anakin was the prophesied one who would bring balance to the Force?
Qui-Gon Doesn't Believe In Accidents
There were a number of factors that led Qui-Gon to believe as he did. The first and main trigger for his suspicion was the chosen one prophecy itself:
"A Chosen One shall come, born of no father, and through him will ultimate balance in the Force be restored."
Qui-Gon had already sensed the Force in the boy. After Anakin insists they come to their home, Qui-Gon catches how perceptive and talented Anakin is beyond his years (Anakin identifies him as a Jedi Knight, talks of how he's the only human who can pod race). Qui-Gon notes as much to Shmi and asks about his father, to which she informs him that Anakin had no father. It's after this that he decides to administer the midi-chlorian test, and confirms his incredible "off the chart" Force potential, which he then sees in action when Anakin wins the pod race. Fate placed this boy in their path, and it was hard to ignore that there was something really special about him.
"Our meeting was not a coincidence. Nothing happens by accident."
While this doesn't necessarily guarantee it, Qui-Gon's faith in the Force and its order in the universe, that nothing happens by accident, and the timing of the appearance of a Sith, lend a strong sense of something big going on, and gives him a strong suspicion that Anakin might be more than he appears.