Oh, yeah you should be totally fine. Justice Points are earned by completing dungeons, yes, but you only really start getting them in a substantial quantity around the current expansion level, so 85 to 90.
Here's a good table to see where you can get them. If you're boosting from 60 to 90, you won't be able to earn any JP unfortunately, and even if you were earning them through Burning Crusade dungeons, it'd take absolutely forever to even get one item.
You can keep an eye on your Justice points through the 'Currency' tab, off the top of my head, I believe you can access them through a tab where you open up your character sheet (C) or Reputations (U).
edit: Yeah, just checked and that's right, if you're 27 and have mainly been questing and doing dungeons, you may not even have the tab 'unlocked' yet, because it's mainly used to store things like Justice/Valor Points, however if you've done PvP or any world events, Darkmoon Faire Prize Tickets are there as well, along with any Honor points you may have earned.