What you are refering to is called in Islamic History as the "Year of the Elephant" where a Yemeni general led his men with the intent to destroy Mecca because of various reasons, my favourite being that someone evacuated their bowels on the governor's shiny new Church he had built.

There is enough evidence that elephants were used. Remember that Ethiopia, Axum at this point, had been allied to the Byzantines in the Sasanian-Byzantine struggle and that Yemen, and coincidentally the Red Sea, had been a source of conflict between the two.

I would not doubt that Elephants were used by the Ethiopian backed Yemeni leader although the evidence as you stated, both in the Quran and in other accounts such as ibn Ishaq, state that only one or two were used and were not in large number.

Since there is no clear description of what it looked like, except being white, I would state that it is more likely referring to an Indian elephant that was imported from either India or Sri Lanka. This is solely due to the fact that a similar white elephant had existed during the battle of al-Qadisiyyah in 636 CE among the Sasanian ranks. The Sasanians are known to have imported their elephants primarily from Sri Lanka thus it is presumed, on my part, that where ever the Sasanians got them the Axumites could get it too.

There isn't much written about this subject except for a couple of primary sources of which the most important I would say is Ibn Ishaq's al-Mubtadaʾ wa al-Baʿth wa al-Maghāzī. The majority of this work is consider lost although the only part that survives is the biography of the Prophet Mohammad. Another important work is al-Tarabi's The History of al-Tabari Volume 5: The Sasanids, the Byzantines, the Lakmids, and Yemen which references certain sections of the battle between the Byzantines and the Sasanians.