You will need to use the PL/PgSQL EXECUTE statement, via a DO block or PL/PgSQL function (CREATE OR REPLACE FUNCTION ... LANGUAGE plpgsql). Dynamic SQL is not supported in the ordinary SQL dialect used by PostgreSQL, only in the procedural PL/PgSQL variant.
DO
$$
BEGIN
EXECUTE format('CREATE TABLE %I AS SELECT * FROM backup', 'backup_' || to_char(CURRENT_DATE,'yyyy-mm-dd'));
END;
$$ LANGUAGE plpgsql;
The format(...) function's %I and %L format-specifiers do proper identifier and literal quoting, respectively.
For literals I recommend using EXECUTE ... USING rather than format(...) with %L, but for identifiers like table/column names the format %I pattern is a nice concise alternative to verbose quote_ident calls.
