Is what Al-Zarqani mentioned in "Manahil Al-Irfan" regarding the writing of the Quran as it became available and its placement in the Prophet's house, may God bless him and grant him peace, correct? And should this be mentioned to a non-Muslim when discussing the truth of Islam?

The existence of a massive library in the house of prophethood, or hundreds of pens and papers, is not a necessary implication. What was present in the house of prophethood cannot be described as a massive library, nor does it necessitate the existence of hundreds of pens and papers. This is because there is no evidence that the Qur'an was fully compiled in the house of the Prophet (peace be upon him).

The Prophet (peace be upon him) passed away, and the Holy Qur'an was not compiled in one place. This is evidenced by the story of the compilation of the Qur'an during the era of Abu Bakr Al-Siddiq (may Allah be pleased with him), which showed that the Qur'an was dispersed among the Companions, and that Zayd ibn Thabit (may Allah be pleased with him) exerted great effort in gathering it from its various sources.

The intent of placing the Qur'an in the house of prophethood was to record what the Prophet (peace be upon him) brought forth, not to author or write it.