If I play a Longsword, which has quick-draw keyword, does the quick-draw equip count as using an activated ability?

No, the Quick-Draw attach does not count as using an activated ability. Quick-Draw is a Triggered Ability keyword (Rule 819.1), while Equip is an Activated Ability keyword (Rule 818.1) — they are mechanically distinct.

When you play Long Sword, you pay only the play cost (2 Energy + 1 Power). Quick-Draw then triggers automatically as a play effect: "When you play this, attach it to a Unit you control" (Rule 819.1.d). This triggered ability places itself on the chain and resolves for free — no Equip cost, no activation of the Equip keyword, and no targeting step on your part beyond the trigger's own choice.

The Equip ability ([C]: Attach this to a unit you control) only comes into play later if the gear becomes unattached and you want to re-attach it. That would be activating an activated ability and would require paying the [C] cost.