I don’t think so, without adding game time and/or resurrecting your account, but I don’t know for sure.
You need to “not be a new player”. I’m not exactly sure what that means. I know that people who have hit level 50 and opened Boralus in BfA can qualify as “not being new”, but I’ve also seen reports that it sometimes doesn’t happen until you have 30 hours /played on that account, even for returning players on existing accounts who have hit 50.
Access to expansions (and the original dungeons are now considered part of the Cataclysm expansion)i by talking to Chromie in Stormwind or Org and accssing “Chromie Time”, allowing you to play through a specific expansion from level 10 to 50.
“New Players” can’t access Chromie Time.
I did meet a level 20 Twink on a Starter account who could access Chromie Time, but I think he had a second, paid account WoW account in his Battle.net account on which he had qualified as “not new”.
You could maybe try going through the BfA quest chain to Boralus. I don’t think that will work, but I don’t know for sure. I’ve seen nothing in announcements or Support that says exactly how to qualify.