|MadSci Network: Astronomy|
The Moon does have "moonquakes", although they occur for different reasons than earthquakes do here on Earth.
Earth's quakes are the result of plate tectonics, the movement of pieces of the Earth's crust. The plates of Earth's crust can move apart from one another, move past one another, or collide with one another. This produces three kinds of earthquakes, extensional, transforms, and compressionals.
The NASA Apollo missions discovered that "there are at least four different kinds of moonquakes: (1) deep moonquakes about 700 km below the surface, probably caused by tides; (2) vibrations from the impact of meteorites; (3) thermal quakes caused by the expansion of the frigid crust when first illuminated by the morning sun after two weeks of deep-freeze lunar night; and (4) shallow moonquakes only 20 or 30 kilometers below the surface". (http://www.nasa.gov/mission_pages/exploration/mmb/15mar_moonquakes.html The Moon is geologically dead today, and probably hasn't had any quakes of tectonic origin for a long time.
Mercury, Venus, or Mars all show geological features that indicate they have had the tectonic activity necessary to produce quakes, but we don't have evidence of quakes there today. This is because we haven't sent the proper instruments to measure seismological activity. The conditions on Mercury and Venus are exceedingly hostile to space probes, and missions to Mars have had higher priorities - like looking for evidence of water. Based on the apparent ages of their geological features, we don't expect Mercury and Mars to be active and have quakes today, except those caused by impacts and, on Mercury, tidal interactions with the Sun and thermal shock. Venus shows indirect evidence for ongoing volcanic and geologic activity, so it probably continues to have "venusquakes".
Jupiter, Saturn, Uranus, and Neptune don't have solid surfaces, so they wouldn't have "earthquakes". The gas giant planets (and the Sun) do have internal oscillations due to pressure waves, which can be studied to learn about the interior of the planet (or the Sun). The vibrations that geologists measure from earthquakes result from seismic waves.
Nearly all the planets discovered beyond our solar system are Jupiter-sized, so they likely would not have "earthquakes" but those oscillations instead. However, it will be a long time (and take many advances in technology) before we can observe them.
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