The account of Uzziah in Josephus' Antiquities of the Jews makes a connection between the earthquake mentioned at the beginning of the Book of Amos and Uzziah being struck down with leprosy after he offered a sacrifice in the temple.
"The words of Amos, who was among the shepherds of Tekoa, which he saw concerning Israel in the days of King Uzziah of Judah and in the days of King Jeroboam son of Joash of Israel, two years before the earthquake." (Amos 1:1)
The earthquake is also mentioned in the Book of Zechariah
"And you shall flee by the valley of the LORD's mountain, for the valley between the mountains shall reach to the Azal; and you shall flee as you fled from the earthquake in the days of king Uzziah of Judah. Then the LORD my God will come, and all the holy ones with him." (Zechariah 14:5)
Josephus: Antiquities of the Jews 9:10:4
While Uzziah was in this state and making preparations (for futurity), he was corrupted
in his mind by pride, and became insolent, and this on account of that abundance which he had
of things that will soon perish, and despised that power which is of eternal duration (which
consisted in piety towards god, and in the observation of his laws); so he fell by occasion of the
good success of his affairs, and was carried headlong into those sins of his father, which the
splendour of that prosperity he enjoyed, and the glorious actions he had done, led him into, while
he was not able to govern himself well about them. Accordingly, when a remarkable day was
come, and a general festival was to be celebrated, he put on the holy garment, and went into the
temple to offer incense to God upon the golden altar, which he was prohibited to do by Azariah
the high priest, who had fourscore priests with him, and who told him that it was not lawful for
him to offer sacrifice, and that "none besides the posterity of Aaron were permitted so to do."
And when they cried out, that he must go out of the temple, and not transgress against God, he
was wroth at them, and threatened to kill them, unless they would hold their peace. In the
meantime, a great earthquake shook the ground, and a rent was made in the temple, and
the bright rays of the sun shone through it, and fell upon the king's face, insomuch that the
leprosy seized upon him immediately; and before the city, at a place called Eroge, half the
mountain broke off from the rest on the west, and rolled itself four furlongs, and stood still
at the east mountain, till the roads, as well as the king's garden, were spoiled by the
obstruction. Now as soon as the priest saw that the king's face was infected with the leprosy,
they told him of the calamity he was under, and commanded that he should go out of the city as
a polluted person. Hereupon he was so confounded at the sad distemper, and sensible that he was
not at liberty to contradict, that he did as he was commanded, and underwent this miserable and
terrible punishment for an intention beyond what befitted a man to have, and for that impiety
against God which was implied therein. So he abode out of the city for some time, and lived a
private life, while his son Jotham took the government; after which he died with grief and anxiety
at what had happened to him, when he had lived sixty-eight years, and reigned of them fifty-two;
and was buried by himself in his own gardens.