Question
Is it allowed for the women to request Mahr to be from the Quran, either all of it or part of it ?
No, absolutely no. The Mahr should be something physically valuable, things like money and its likes. So the Mahr of the women could be money, cars or buildings that are equal to money to benefit her now or later. And this is how the Mahr should be and this is how Allah عز وجل mentioned the Mahr in the Quran and Muhammad ﷺ mentioned in his Sunnah. If you see all the evidences from the Quran and the Sunnah related to Mahr, you see it mentioned as something that is valuable. Like the statement of Allah عز وجل :
فَلَا تَأۡخُذُوا۟ مِنۡهُ شَیۡـًٔاۚ أَتَأۡخُذُونَهُۥ بُهۡتَـٰنࣰا وَإِثۡمࣰا مُّبِینࣰا
" .... take not the least bit of it ( Mahr ) back; would you take it wrongfully without a right and (with) a manifest sin? " [4:20]
وَكَیۡفَ تَأۡخُذُونَهُۥ وَقَدۡ أَفۡضَىٰ بَعۡضُكُمۡ إِلَىٰ بَعۡضࣲ وَأَخَذۡنَ مِنكُم مِّیثَـٰقًا غَلِیظࣰا
" And how could you take it (back) while you have gone in unto each other, and they have taken from you a firm and strong covenant ? " [4:21]
Things that you give and return usually means something (physically) valuable.
And Allah عز وجل said :
وَقَدْ فَرَضْتُمْ لَهُنَّ فَرِيضَةً فَنِصْفُ مَا فَرَضْتُمْ
" And you have appointed unto them the Mahr, then pay half of that (Mahr) " [2:237]
This verse necessitates that the Mahr here is something that can be divided into two shares. This no doubt is applicable only for things that are physically valuable like money.
And similar things are mentioned in the Sunnah. And this is the opinion of the majority of the Ulema from the Salaf and the recent scholars too. All of them describe the Mahr as something that is valuable like money and other properties equivalent to money. Because of this, majority of the Ulema including all the four Madhahib, didn't allow that the Mahr can be something that does not have any physical value such as the Mushaf or Sahih Bukhari.
And if someone mentions the Hadith of Sahl bin Sa'd رضي الله عنه in the Saheeh that the Prophet صلى الله عليه وسلم married a woman to a man with what he has from the Quran as Mahr then that is because he did not possess anything other than that. The evidence for this is that, the Prophet صلى الله عليه وسلم told the man to go and seek anything that is valuable until the Prophet صلى الله عليه وسلم said " Go and search even for a ring made from metal " which is the least value that a poor person can afford. But he couldn't afford even that so the Prophet صلى الله عليه وسلم allowed him to marry with what he had from the Quran meaning he had to teach her from the Quran.
And regarding the Hadith of Umm Sulaym رضي الله عنها and how she made a condition for Abu Talha to marry her he should accept Islam, then this was during the beginning stages of Islam since Abu Talha became a Muslim during the early days amongst the Ansar. And the verses that mention that the Mahr should be something physically valuable, they were only revealed to the Prophet صلى الله عليه وسلم during the end of his life and these verses are from the later Suwar (Surat an Nisa) of the Quran.
And also the scholars say that the Mahr was infact paid to Umm Sulaym رضي الله عنها even if it was not mentioned explicitly since it was from their culture and lifestyle. The condition that Abu Talha رضي الله عنه should become a Muslim was an extra condition besides the normal Mahr. Also it is well known that Abu Talha رضي الله عنه was from the rich ones from the Ansar رضي الله عنهم so why should he not pay the Mahr..?
So I advise you my brothers and sisters that you have to follow the path of the Ulema and the correct understanding of the Salaf regarding this issue and give a proper Mahr to your spouse. The Ulema also said that if someone didn't give a proper Mahr to his wife because of a wrong understanding concerning the issue, he must pay her Mahr properly again once he has realised his mistake even if it is after many years.. it is still obligatory upon him. Even though the Nikah is valid and correct he still need to pay it. We say the same regarding some Indians and Pakistanees whose tradition is that their woman has to pay the dowry to the man. This is not allowed even though the Nikah is still valid. However it is obligatory for the husband to back the Mahr.
And Allah عز وجل knows best.