Checking from the package formula postgresql.rb by brew cat postgresql, the datadir is ${HOMEBREW_PREFIX}/var/postgres. (HOMEBREW_PREFIX is not an exported env var, it's used by homebrew internally)
- /usr/local/var/postgreson x86 Mac
- /opt/homebrew/var/postgreson M1 Mac
This dir should be created automatically during brew install postgresql.
BTW, the canonical way to start psql with Homebrew should be brew services start postgresql, which uses launchd on macOS to manage services. (For newbie on macOS, launchd could be seen as a systemd alternative on macOS.)
Run brew info postgresql and some additional tips is displayed in the Caveats part of the output.
❯ brew info postgresql
postgresql: stable 13.3 (bottled), HEAD
Object-relational database system
https://www.postgresql.org/
/usr/local/Cellar/postgresql/13.3 (3,225 files, 38.8MB) *
  Poured from bottle on 2021-05-14 at 09:37:24
From: https://github.com/Homebrew/homebrew-core/blob/HEAD/Formula/postgresql.rb
License: PostgreSQL
==> Dependencies
Build: pkg-config ✔
Required: icu4c ✔, krb5 ✔, openssl@1.1 ✔, readline ✔
==> Options
--HEAD
        Install HEAD version
==> Caveats
To migrate existing data from a previous major version of PostgreSQL run:
  brew postgresql-upgrade-database
This formula has created a default database cluster with:
  initdb --locale=C -E UTF-8 /usr/local/var/postgres
For more details, read:
  https://www.postgresql.org/docs/13/app-initdb.html
To have launchd start postgresql now and restart at login:
  brew services start postgresql
Or, if you don't want/need a background service you can just run:
  pg_ctl -D /usr/local/var/postgres start