The quest Guidance of Apollo only requires that the player find the Solar coin. After they do, their notebook will say:
"You found the Solar Coin. It seems heavier and more beautiful than any of the other coins. Take another look at it in the collection book."
As you do so, the strange voice from the radio will make an appearance, stating:
"...the one...who rules the sun...find the coin...cavern...with the same name...inside the cliff...inherit...riches of the north..."
Your notebook will update:
"You heard a strange voice from the radio. It seemed to say that there's something in "the cavern that shares the coin's name"."
When you go to the Cavern of the Sun, you will find the Turtle shell Tank, a unique piece of equipment. Your notebook will update with this piece of information, noting "You found something rare in the Cavern of the Sun."