Protection works by checking whether the source of damage has the quality the protection names. Protection from multicolored means the protected permanent cannot be damaged by, enchanted/equipped/fortified by, blocked by, or targeted by anything that is multicolored (CR 702.16b).
A card is multicolored if it has two or more colors (CR 105.2c). A card is colorless only if it has no colors at all (CR 105.2). These two properties are mutually exclusive — you cannot simultaneously have multiple colors and no colors. Therefore, a 'gold card' (a card with two or more colors) is inherently not colorless by definition.
If a card started as multicolored but somehow lost all its colors (e.g., via an effect that makes it colorless), it would no longer be multicolored, so protection from multicolored would not apply to it — but that same effect would also make it no longer a 'gold card' in the traditional sense.
Example: A creature with protection from multicolored is attacked. The attacking creature is a Boros Charm token with white and red mana symbols in its cost — it is multicolored. The protection prevents all damage from that attacker. There is no realistic scenario where a card is simultaneously gold (multicolored) and colorless under the rules.
In short: the premise of the question describes an impossible game state under the rules. Any card that is truly multicolored cannot also be colorless, so protection from multicolored will always apply to a multicolored source regardless of any confusion about 'gold' terminology.
Unofficial fan resource — not affiliated with or endorsed by Wizards of the Coast. Answers are AI-generated estimates grounded in the Comprehensive Rules and are not a substitute for an official judge. Verify anything match-critical.