In Norse Mythology
Fafnir was the son of the dwarf king Hreidmar. One day, Odin, Loki and Hoenir accidentally murdered Otr, who had taken the form of an otter. When Hreidmar learned of this, he captured the three Aesir with the help of Fafnir and his brother Regin. They held them for ransom, allowing Loki to leave so he could pay. Loki offered the dwarves Andvari's ring and his cursed gold, which gave Fafnir the inspiration to murder Hreidmar. Due to his obsession with the gold, Fafnir slowly became a dragon, which is a symbol of greed. He also possessed poisonous breath, which he shrouded himself in so that no one would try to steal his treasure. Eventually, Regin, corrupted by greed, sent his apprentice Sigurd to kill Fafnir to reclaim the gold. After Fafnir had been mortally wounded by his adopted nephew, he warned Sigurd that all who claimed the gold would die somehow.
In God of War
Fafnir was a Dwarf, like Brok and Sindri, he was known, even by Mimir to travel to different realms for items that interest him. The Whetstone that Sindri sends Kratos and Atreus to retrieve was Fafnir's, where he once ventured into Niflheim for it, which Sindri says "Must have been some Whetstone".
In Fafnir's storeroom he had many a treasure from all realms, however, a group of Reavers stole nearly all treasure, minus one chest.
Fafnir was soon discovered by Kratos and Atreus who was imprisoned like the other dragons, Atreus is surprised by Mimir's comment that this dragon was Fafnir. Mótsognir must have found him and captured him, in his own former home no less, to create a legendary dwarf armor. A lore marker soon reveals that he had taken a Vanir mirror that when glanced at could turn one into a Dragon. After being freed, Mimir remarks that being turned into a dragon didn't really improve his terrible attitude.