Can You Put the Quran on the Prayer Mat?
Question

What is the ruling on putting the Quran on the prayer mat?
Praise be to Allah.There is no difference of opinion among the scholars concerning the fact that the Quran must be respected and protected. 
Al-Nawawi (may Allah have mercy on him) said: 
“The scholars unanimously agreed that it is obligatory to protect and respect the Mus-haf.” (Al-Majmu`, 2/85) 
The Muslim should not go to extremes in respecting the Quran. People have gone to extremes in this matter and have made it too burdensome, as it was narrated that one of them said: “For thirty years I have not entered a room in which there is a Mus-haf without having Wudu.” And if one of them is in a room in which there is a Mus-haf he would not sleep all night lest he break wind in a room in which there is a Mus-haf !  
These actions are clearly contrary to the way of the Prophet (peace and blessings of Allah be upon him) and his Companions (may Allah be pleased with them). They lived in small, cramped rooms, but that did not prevent them from sleeping in their houses, or having intercourse with their wives , or staying without Wudu for a while, even though there were Mus-hafs in their houses. When the Quran was collected, it was kept in the houses of many of them. 
There were no shelves in the mosque of the Prophet (peace and blessings of Allah be upon him) or the mosques of the Companions on which to place the pages of the Mus-hafs, and papers containing knowledge. Hence what matters here is whether this is regarded as disrespectful or not. Hence there is nothing wrong with putting it on clean ground for one who needs to do that – such as if he wants to do the prostration of recitation (Sujud At-tilawah ). 
Shaykh Muhammad ibn Salih Al-`Uthaymin (may Allah have mercy on him) said: 
“Sincerity towards the Book of Allah includes not putting it in places where it will be treated with disrespect, or where putting it there is a kind of disrespect towards it, such as a place where garbage is put and the like. Hence it is essential to avoid doing what some children do when they finish their studies, when they throw their books which may contain part of the Quran in the streets or in the garbage and so on. Allah forbid. 
As for putting the Quran on clean ground, there is nothing wrong with that, because this is not disrespecting the Quran or mistreating it. It often happens that a person is praying or reading from the Mushaf and he wants to prostrate, so he puts it down in front of him. This is not regarded as disrespect or mistreatment of the Mus-haf, so there is nothing wrong with it.” (Sharh Riyadh As-Salihin (1/423), Dar Ibn Al-Haytham, Hadith no. 181) 
Shaykh `Abdullah ibn Jibrin (may Allah preserve him) was asked: 
What is the ruling on putting the Quran on the clean ground or on the prayer mat? 
He replied: 
“It is better to put it in a high place so that it will be exalted in both the literal and metaphorical sense. Allah says (interpretation of the meaning): “Exalted (in dignity), purified” [‘Abas 80:14]. If you need to put it down, then put it down in a place that is raised up, even if only a little. If that is not possible then it is permissible to put it on the ground, on a clean carpet and the like. The Quran should not be put down in a low place or a place that is impure or on the dirt, because that is showing disrespect towards it. But if a person needs to put it down on a clean carpet, there is nothing wrong with that, so long as one is keen to exalt it both literally and metaphorically.” (Fatawa Islamiyyah, 4/15) 
Based on that, if the mat is clean and it is unlikely that children or others will mess about with the Mus-haf , there is nothing wrong with putting the Mus-haf on it, but putting it in a higher place is better. 
And Allah knows best.