Shutting up certainly sounds like a good idea, at first. However, think about what happens next: specifically, how do you convert the rare antique gold coins to cash without drawing attention. Worse, once you are asked where the coins came from, you may face criminal or civil penalties for not reporting them in the first place. At that point, you are worse off than before you found them.
You talk as if it’s Plutonium. Gold coins are the most liquid asset in the world, not quite as good as cash. Go to coin dealer. “How much?” Agree on price. Takes about 2 minutes. Less if you aren’t talkative.