According to Josephus:
Ezekiel also foretold in Babylon what calamities were coming upon the people, which when he heard, he sent accounts of them unto Jerusalem. But Zedekiah did not believe their prophecies, for the reason following: It happened that the two prophets agreed with one another in what they said as in all other things, that the city should be taken, and Zedekiah himself should be taken captive; but Ezekiel disagreed with [Jeremiah], and said that Zedekiah should not see Babylon, while Jeremiah said to him, that the king of Babylon should carry him away thither in bonds.—The Antiquities of the Jews, 10.106 (translated by William Whiston, 1737).
I wanted to find out where Ezekiel disagreed with Jeremiah. The passage seems to be:
Say, ‘I am a sign for you: as I have done, so shall it be done to them. They shall go into exile, into captivity.’ And the prince who is among them shall lift his baggage upon his shoulder at dusk, and shall go out. They shall dig through the wall to bring him out through it. He shall cover his face, that he may not see the land with his eyes. And I will spread my net over him, and he shall be taken in my snare. And I will bring him to Babylon, the land of the Chaldeans, yet he shall not see it, and he shall die there.—Ezekiel 12:11-13 (ESV)
Now it's true that Ezekiel said that Zedekiah would not see Babylon, but it's clear from 2nd Kings 25:7 and Jeremiah 52:11 that the reason he won't see Babylon is because Nebuchadnezzar II put out his eyes. Reading Ezekiel the situation is a bit confused, but it seems clear that he is prophesying that Zedekiah would come to a bad end in Babylon. There seems no contradiction with Jeremiah at all.
Did Josephus misread Ezekiel or did he have access to a copy of the text that did not clearly predict that Zedekiah would be taken into captivity? Is there some other place where Ezekiel disagreed with Jeremiah more clearly?
Josephus seems to be referring to this passage in Jeremiah that says Zedekiah would be taken to Babylon:
“Thus says the LORD, the God of Israel: Go and speak to Zedekiah king of Judah and say to him, ‘Thus says the LORD: Behold, I am giving this city into the hand of the king of Babylon, and he shall burn it with fire. You shall not escape from his hand but shall surely be captured and delivered into his hand. You shall see the king of Babylon eye to eye and speak with him face to face. And you shall go to Babylon.’—Jeremiah 34:2-3 (ESV)