This happens in Chapter 5, "Riddles in the Dark", when Bilbo and Gollum are playing a game of riddles in the dark caves beneath the Misty Mountains. Gollum, frustrated by Bilbo's cleverness, tries to force him to answer a riddle about what has no voice but can speak. Bilbo, realizing the answer is "the river", shouts it out and in doing so, accidentally stumbles into the stream.
This moment is significant because it marks the first time Bilbo shows any courage and resourcefulness on his journey.