After delivering Gerdur's message to Jarl of Whiterun and convincing him to send forces to protect Riverwood from the dragon, Jarl asks you for a favor.
Bleak Falls Barrow
Jarl Balgruuf thinks I may be able to help Farengar, his court wizard, with something related to dragons.
Talk to Farengar
Retrieve the Dragonstone
Deliver the Dragonstone to Farengar
Jarl will lead the way to his court wizard, follow him into the room nearby. Jarl will tell Farengar that he thinks he has found someone who can help the wizard with his dragon project and to fill you in with all the details. Farengar tells you about his dragon research. He needs someone to fetch something for him. An ancient stone tablet from some buried ruins. He has learned of a Dragonstone, an ancient tablet, buried inside Bleak Falls Barrow, in the main chamber. The stone contains a map to dragon burial sites. If you check your quest journal, you should see it has been updated:
Farengar Secret-Fire, Jarl Balgruuf's court wizard, has asked me to retreive a Dragonstone from Bleak Falls Barrow, a stone tablet that he believes contains a map of ancient dragon burial sites. He hopes this will help him understand how and why dragons are returning to Tamrial
IMPORTANT: This quest should be done together with the "The Golden Claw" quest that you get from Lucan Valerius in Riverwood, since both quests take you to the same area in Bleak Falls Barrow. Go to the Riverwood Trader and get that quest from the shopkeeper there. If you already completed "The Golden Claw", then you should already have the Dragonstone on you! Just hand it over to Farengar to complete the quest.
CLICK HERE FOR THE WALKTHROUGH OF THE GOLDEN CLAW QUEST
After retrieving the dragonstone, give it to Farengar. This will complete the quest, and unlock the next quest in the campaign, "Dragon Rising".