From what I understand it was 587 BCE, but my Jehovah’s Witness mother refuses to beleive otherwise and thinks it’s 607 BCE.
Is there any proof of one over the other?
Apparently this is a somewhat common theological debate because I've answered almost the exact same question before. You'll see that there's not just one "fall" of Jerusalem to Babylon.
That answer doesn't address 607 BCE specifically because it's not a date that comes up directly in reference to Jerusalem. Where the JWs or just your mother got the date I don't know. It's actually a weird date to choose, even the scheme of common Christian misconceptions of the dates associated with the events. 609 would make sense because that's when Babylon formally conquered the Assyrian Empire. 605 would make sense because that's when Babylon had its major victory against the Egyptians in the Levant at Karchemish (who killed King Josiah and subjugated Jerusalem on their way north in 609 according to 2 Kings and 2 Chronicles). None of those dates actually make sense to "fall" to Nebuchadnezzar because his father, Nabopolassar, only died in 604 BCE.