Are there earthquakes on planets other than Earth?
The easy answer is “no”. They would not be called “earthquakes” if they don’t occur on Earth. The real answer is that we would expect earthquakes to occur on any rocky planet which has Plate Tectonics. In our Solar System, only the Earth has active Plate Tectonics. But other solar systems will have formed in the same way as ours and it would be very surprising if there were no planets with Plate Tectonics.
We detect earthquakes on Earth using sensitive instruments called seismometers. Four of the Apollo missions to the Moon deployed seismometers. You can find a photo and some more information on NASA’s web site. There is no evidence of Plate Tectonics on the Moon and so scientists were a bit surprised at the number of Moonquakes they recorded. Most of these are generated by the strong gravitational force of the Earth. NASA are now planning to return to the moon and might have to plan for moonquakes.
One of the early unmanned missions to Mars (Viking 2) included a seismometer. In the short time it was operating, it recorded no conclusive evidence of Marsquakes. None of the recent Mars missions included seismometers.
Amazingly, there is some evidence for earthquakes on the Sun, or Sunquakes. These Sunquakes are very different from Earthquakes and Moonquakes and maybe should be called something else.