Based on the Jax Unmatched errata, can you attach an unattached equipment on the field as Quickdraw?

No, you cannot use Quick-Draw to attach an Equipment that is already on the battlefield.

According to RiftJudge FAQ #9085 and the definition of the keyword, Quick-Draw only functions when you play an Equipment card from your hand.

Here is the breakdown of why this is the case:

1. Quick-Draw Timing: As defined in Rule 819.1.d and clarified in the FAQs, Quick-Draw is functionally "[Reaction]" and "When you play this, attach it to a Unit you control." This effect only triggers when the card is moved from your hand to the battlefield (played).
2. Existing Gear: An Equipment already on the battlefield has already been played. Because it is no longer being "played" from your hand, the Quick-Draw triggered ability does not activate.
3. Equip vs. Quick-Draw: The Equip ability (Rule 818) is a separate activated ability used to move Gear that is already on the board. As clarified in RiftJudge FAQ #9085, Jax does not change the timing restrictions for activating an existing Equipment's Equip ability; you must still follow standard timing rules (Rule 148.2), which restrict it to your own Main Phase during a Neutral Open state.

Summary: Quick-Draw is not a universal "attach for free" ability; it is specifically a triggered ability that occurs only when you first play an Equipment from your hand. Once the gear is on the board, you must use the Equip activated ability (and pay its cost) to attach it to a unit.