Question: Sublime Text has this same functionality via:

`ctrl+shift+m` or `cmd+shift+space`

How do I accomplish the same thing in Atom?

Answer: 
Select all text between brackets is supported by default [Bracket Matcher](https://atom.io/packages/bracket-matcher) package.

Works for `{}`, `[]` and `()`:

`Ctrl`+`Cmd`+`M`

To select text inside quotes, you can install [expand-selection-to-quotes](https://atom.io/packages/expand-selection-to-quotes) package and use:

`Ctrl`+`'`
Atom does not natively support this functionality that I have found.
The closest thing I have seen is this package: <https://atom.io/packages/select-scope>