No, you should not write it. .For various reasons.
1. You need Wudhu ! Each time you use the Quran and also when you handle your paper. Because it is the Mushaf. (the original Arabic Quranic script).
Non-Muslims might not handle it properly.
When you write the verses, then it must be exactly as in the Quran. So, you must pay attention to the way the letters are. It will not be good if you made mistakes.
It is quite a vast area. It is not as simple as we might think, this is what people with experience and knowledge do. For, they will know how the old Kufic scripts were made, or the ones in North Africa etc are...... Non Muslims may ask you questions which you may not know answer to, such as why in the early versions of the script there no extra fatha, damma, kasra markings etc. Of course, the Quran was always read in exact same way. Just the way the words were written is different. Just as someone can write a word joint up, in capital letters or each letter seperately. The meaning and pronounciation is same. Of course, the Quran is much more than that and cant be compared to any other text as it is from Allah alone and not the word of man.
I think you should look into doing something else, Arabic writing has changed a lot over time and in different ways depending on region. The Quran has different styles of recitation, such as the Warsh in Morocco or the Qalun in Libya, and thus writing is different. It may not be easy for non-Muslims or people who dont have much knowledge this subject to understand.
You could do something about change in Islamic art, such as mathematics of tesselating patterns etc.. lol.. or something bit more simple is the difference of art in Islamic countries, Xinjiang has its own Chinese style Islamic art, Morocco has its own, Iran, Syria its own, Turkey its own.. what are the similarities? what are the differences ? So, there is quite a bit you can research on that.
EDIT... you can do your own name, inshaAllah.. and try and write it in different ways... you can do in two ways, 1. maybe chart it on timeline, so people can see how it might have changed according to time. or you can do it according to the Region, how it may have been written in Iraq , how it may have been written in Egypt or Morocco, you can put your art work on map of world or something.
Wasalam.
