As a follow up to my post about this happening, I got an email from SE support about this finally.

"Thank you for taking the time to alert us to "abandoning a quest", and erasing the progress you made towards its completion. Any loss of progress is a frustrating experience, since that involves not just gil, but time and effort that has been lost. At the current time, the GMs and Support Staff are unable to restore quest progress; however, due to reports like this, we are currently looking into how we may be able to assist with these issues. We apologize for the inconvenience and that we do not have an immediate resolution that we can offer you. If you feel that abandoning the quest was due to a game error and not an accidental choice, please report it through the forums so we can track the progress of the report. Thank you again for your patience and understanding. Final Fantasy XIV Support Team"
Well then, seeing the subject of the email was "Template", I'll assume you didn't even read the original message I sent in saying the quest was dropped without me "accidently" doing it. I exited the garuda hard mode duty, the quest was gone. It's that simple.