A usual term is "scribe" as to those who were appointed by Prophet Mohammad to write down the verses of Quran as and when they were revealed to him. So are you asking for the names of these scribes or what?  

In the Name of Allah, the Most Beneficent, the Most Merciful

Peace and blessings of Allah be upon Prophet Muhammad, his family and his companions

Assalamu Alaykum Wa Rahmatullahi Wa Barakatuh

Insha Allah, you will find the answer to your question below:

(Taken from http://www.witness-pioneer.net )

Brief History of Compilation of the Qur'an
Adapted from an article in Perspectives, Vol 3, No. 4, Aug/Sept 1997
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During the life of the Prophet (saas) (570-632 CE)

-The Prophet (peace & blessings of Allah be upon him) used to recite the Qur'an before angel Jibreel (Gabriel) once every Ramadan, but he recited it twice (in the same order we have today) in the last Ramadan before his death. Jibreel also taught the Prophet ( Peace and blessings of Allah be upon him) the seven modes of recitation.

-Each verse received was recited by the Prophet (Peace and blessings of Allah be upon him), and its location relative to other verses and surahs was identified by him.

-The verses were written by scribes, selected by the Prophet (Peace and blessings of Allah be upon him), on any suitable object - the leaves of trees, pieces of wood, parchment or leather, flat stones, and shoulder blades. Scribes included Ali Ibn Abi Talib, Mu'awiyah Ibn Abi Sufyan, Ubey Ibn Ka'ab, Zayed Ibn Thabit.

-Some of the companions wrote the Qur'an for their own use.

-Several hundred companions memorized the Qur'an by heart.

During the caliphate of Abu Bakr (632-634 CE)

-Umar Ibn Al-Khattab urged Abu Bakr to preserve and compile the Qur'an. This was prompted after the battle of Yamamah, where heavy casualties were suffered among the reciters who memorized the Qur'an.

-Abu Bakr entrusted Zayed Ibn Thabit with the task of collecting the Qur'an. Zayed had been present during the last recitation of the Qur'an by the Prophet to Angel Jibreel (Gabriel).

-Zayed, with the help of the companions who memorized and wrote verses of the Qur'an, accomplished the task and handed Abu Bakr the first authenticated copy of the Qur'an. The copy was kept in the residence of Hafsah, daughter of Umar and wife of the Prophet (Peace and blessings of Allah be upon him)..

During the caliphate of Uthman (644-656 CE)

-Uthman ordered Zayed Ibn Thabit, Abdullah Ibn Al Zubayr, Saeed Ibn Al-Aas, and Abdur-Rahman Ibn Harith Ibn Hisham to make perfect copies of the authenticated copy kept with Hafsa. This was due to the rapid expansion of the Islamic state and concern about differences in recitation.

-Copies were sent to various places in the Muslim world. The original copy was returned to Hafsa, and a copy was kept in Madinah.

Three stages of dotting and diacritization

-Dots were put as syntactical marks by Abu Al-Aswad Al Doaly, during the time of Mu'awiya Ibn Abi Sufian (661-680 CE).

-The letters were marked with different dotting by Nasr Ibn Asem and Hayy ibn Ya'amor, during the time of Abd Al-Malek Ibn Marawan (685-705 CE).

-A complete system of diacritical marks (damma, fataha, kasra) was invented by Al Khaleel Ibn Ahmad Al Faraheedy (d. 786 CE).

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May I suggest something to you "beloved"! Although it is absoultely correct to ask questions about Islamic issues which you have no knowledge about, it would be more safe on your part to ask questions in a way which do not imply anything offensive. In my opinion, your first question implied "doubts" in regard to the authenticity of the Holy Quran! It is not fitting for a muslim to doubt anything about the Holy Quran. So, if you are a muslim, please ponder over the following Verses of the Noble Quran to know the quality of true believers: 

002 - Al-Baqarah   285. The Messenger (Muhammad SAW) believes in what has been sent down to him from his Lord, and (so do) the believers. Each one believes in Allah, His Angels, His Books, and His Messengers. They say, "We make no distinction between one another of His Messengers" - and they say, "We hear, and we obey. (We seek) Your Forgiveness, our Lord, and to You is the return (of all)."

024 - An-Nur   51. The only saying of the faithful believers, when they are called to Allah (His Words, the Quran) and His Messenger (SAW), to judge between them, is that they say: "We hear and we obey." And such are the prosperous ones (who will live forever in Paradise).

May Allah guide us to His Straight Way.

Bismillah

To make things a little clearer may I suggest using the rightly mentioned word "scribes" instead of "writers". It is a good attempt to confuse readers, but fortunately these matters are well taken care of in Islam... as opposed to Christianity.

Also, is it safe to say that even though the Quran was written down by different people (the above mentioned "scribes"), they all did indeed write down the same material? So it is not like the Gospel which was written differently by an undetermined number of unknown people.