In order to answer your question, one may direct you to this quote (or the similar ones mentioned in the wiki page) to convince you:
Nuru'd-din Bin Sabbagh Maliki in his Fusulu'l- Muhimma, Fasl I, p.14:
"No one before Ali was born in the Ka'ba. This was a distinction given to Ali in order to enhance his honor, rank, and dignity."
[People reported to be born in the Kaaba]
Or one may mention the last dialogues between Sa'sa'a bin Sohan and Imam Ali (AS) before Imam's death:
Sa'sa'a said to Ali: "Let me know who is superior, you or Adam." The Holy Imam said: "It is not proper for a man to praise himself, but according to the maxim: 'Declare the blessings that Allah has given you,' I tell you that I am superior to Adam." When asked why this was so, Ali explained that Adam had every means of mercy, comfort, and blessings for him in paradise. He was asked simply to abstain from the forbidden food. But he could not restrain himself, and he ate from the tree. As a result, he was expelled from paradise. Allah did not forbid him, Ali, from eating wheat (which, according to Muslim belief was the forbidden 'tree'). But since he had no inclination towards this temporal world, he voluntarily refrained from eating wheat. (The point of Ali's remark was that excellence of a man before Allah lies in piety and devotion, and that the height of piety lies in abstaining even from what is permissible.)
[...]Sa'sa'a asked: "Who is superior, you or Jesus?" Ali said: "I am superior, for when Mary became pregnant by the Grace of Allah, and the time of her delivery approached, a revelation was granted to her: 'Leave this holy House for this is a House for prayers, not a place for the delivery of children.' Accordingly, she left the holy House and went to the wilderness where she gave birth to Jesus. But when my mother, Fatima Bint-e-Asad, felt labor pains within the precincts of the holy Ka'ba, she clung to the wall and prayed to Allah in the name of that House and the builder of that House, to lessen her pain. Soon a fissure appeared in the wall, and my mother heard a mysterious voice telling her, "O Fatima! Enter the House of the Ka'ba.' She went in, and I was born inside of the holy Ka'ba."
The Book: PESHAWAR NIGHTS
By: Sultanu'l-Wa'izin Shirazi
Translated by: Hamid Quinlan and Charles Ali Campbell
Published 1996 by Pak Books, P.O. Box EE, Palisades, NY 10964
Or even one may point at the crack on the wall of Kaaba, which is on the other(opposite) side of al-Hajar al-Aswad (the black stone) and unfortunately our Muslim brothers don't let us take photo of it.
But I just want to ask this question from those ones who deny the birth of Imam Ali(Alaihis Salaam) inside Kaaba:
If Imam Ali(AS) was not born inside Kaaba then would you tell me where else was he born? And provide me authenticate documents about it?