In the Star Trek Original series, what episodes did they encounter entities with Q like characteristics?
Restricting ourselves to those who have inexplicable "godlike" powers and an incorporeal form, there are a relatively small number of TOS episodes that fit the description. The most obviously comparable (with Q) are the Organians and Trelane
In addition to the other excellent answer, I'd like to add the two episodes
- Day of the Dove - in which an unnamed non corporeal entity forces the crew to fight some Klingons. Displays the ability to manipulate matter and energy
- And The Children Shall Lead - contains the character Gorgon. A non corporeal being with mind control powers.
Neither of these creatures are up there with Q in terms of power, but both appear in non-canon works with Q, in which the creature from Day of the Dove is given the name (*).