Will my query continue to execute after I shut off my computer
Probably not.

The server will detect that the TCP connection has been closed sooner or later and then try to cancel the query.

You may have success with suspend instead of shutdown, but it's not guaranteed to work.

Your best option is to connect to the server (or to some other always-on server) and run the query there using the psql command-line in a  tmux, or screen session etc.

PgAgent may also be worth considering.