Does protection prevent damage from a source that shares that protection's quality if the source is a triggered ability?

Protection has four functions, remembered by the acronym DEBT: it prevents Damage, stops Enchanting/Equipping, blocks Blocking, and stops Targeting — all from sources that share the protected quality (CR 702.16b). Crucially, the rule applies to damage dealt by a source, and 'source' includes triggered abilities that deal damage.

When a triggered ability deals damage, its source is the permanent or spell that generated it (CR 609.7). So if a creature has protection from red, and a red permanent has a triggered ability that says 'deal 2 damage to target creature,' the source of that damage is the red permanent. Because the source is red, protection from red prevents all that damage (CR 702.16b).

This is true even if the triggered ability is on the stack as a separate object — the damage it deals is still 'from' the original red source. The game always traces damage back to the object that created the ability (CR 609.7a).

Concrete example: Your opponent controls a Chandra's Incinerator (a red creature) whose triggered ability deals 6 damage to your creature. If your creature has protection from red, all 6 damage is prevented, because the source of the damage — Chandra's Incinerator — is red.

One caveat: if the protection is from a specific named card or color word, it must match the source's characteristic at the time damage is dealt — but for typical color or type protections, the triggered-ability damage is fully prevented (CR 702.16b, 609.7).