No, the automatic federal extensions you are referencing are based on the winter storms of January 8 through 10, 2023. Santa Clara is an affected county, but you would not be covered if you moved prior to January 8. Click here for details.
The California extension would apply as they are allowing an extension for all California tax returns.
If you need more time to file your federal tax return, you can get an automatic extension by filing Form 4868. Click here to file an extension in TurboTax.