I don’t believe there’s a correlation. However, once when I was living in the LA area, a friend of mine who grew up in the valley said the weather one night felt like earthquake weather. The next morning, we had a 6.1.
That was the only time I ever heard him say that. Coincidence...
While atmosphere itself isn’t a huge variable in surface loading it is a variable. Tidal action is a lot more predictive of tectonic activity, and both are going to be associated with whichever particular fault system motion loading and time since previous release you’re looking at.
An inch (Hg) change of atmospheric pressure is going to relieve or impart about 3.14 million pounds of load per acre. Each foot of soil is equal to about 5 million pounds per acre (increasing with depth generally and higher for rock).
It’s pretty straight forward that, probability wise, you’ve got a better window of opportunity for releases around lunar cycles and extreme atmospheric events.
Now, resolving evidentiary signs of that out from among other stuff like drought/excess rain, groundwater (or other fluids) extraction or injection, etc... it might get fairly “noisy”.