The activated ability doesn’t target the creature the Equipment will be attached to. It may be attached to a creature with shroud, for example. You don’t choose which creature it will be attached to until you actually attach it. Once you choose which creature to attach it to, it’s too late for players to respond.

If you cast a creature spell, Quest for the Holy Relic’s triggered ability triggers and goes on the stack on top of it. The triggered ability will resolve before the spell does.

You may activate Quest for the Holy Relic’s activated ability even if you control no creatures, or you control no creatures that can be equipped. In that case, you’ll simply put the Equipment card you find onto the battlefield without attaching it to anything.