No. In all of the show's 571 episodes, Timmy (or Lassie's previous owner Jeff) has never fallen into a well once. He was trapped in mines and pipes and many other things, but not a well; in fact, it was Lassie herself who was trapped in a well in a season 17 two-part episode. From Wiki:
"Timmy's in the well!" became a popular catchphrase years after the show's cancellation in reference to the many show situations in which Timmy's safety and welfare were placed in jeopardy. Although Timmy never fell into a well on the show, Jon Provost [who played Timmy] chose the phrase as the title for his 2007 memoirs, Timmy's in the Well: The Jon Provost Story. Provost points out that Timmy fell into abandoned mine shafts, off cliffs, into rivers, lakes and quicksand, but never fell into a well.
As for the origin of this misconception, it's hard to say. Kids fall into wells in real life, and people may have just assumed it happened in Lassie too. It's also possible that they confused mineshafts or pipes for wells, or misremembered the episode where Lassie fell down a well, and the mistake stuck.