In the 4th Century B.C., Aristotle proposed that earthquakes were caused by winds trapped in subterranean caves. Small tremors were thought to have been caused by air pushing on the cavern roofs, and large ones by the air breaking the surface. This theory lead to a belief in earthquake weather, that because a large amount of air was trapped underground, the weather would be hot and calm before an earthquake. A later theory stated that earthquakes occurred in calm, cloudy conditions, and were usually preceded by strong winds, fireballs, and meteors.
There is no such thing as "earthquake weather". Statistically, there is approximately an equal distribution of earthquakes in cold weather, hot weather, rainy weather, etc. Very large low-pressure changes associated with major storm systems (typhoons, hurricanes, etc) are known to trigger episodes of fault slip (slow earthquakes) in the Earth’s crust and may also play a role in triggering some damaging earthquakes. However, the numbers are small and are not statistically significant.