Praise be to Allah.
It is the right of authors of books and programs, for which they have gone to the effort of producing and publishing them, that the copyright should be protected, and they may prevent others from publishing, printing or producing this material except with their permission, as we have explained in the answers to questions no. 95173, 454, 21899 and 26307.
The Mushaf is the book of Allah, may He be exalted, and no one has the right to claim copyright on publishing, printing and distributing it, as far as the content is concerned, unless he has put effort into that which means that he is entitled to copyright, such as one who prints it with a commentary (tafseer) in the margins or the meanings of the words, or one who publishes it in a program for which he has expended money and effort. In such cases, their rights are protected, as we have explained in the answer to question no. 126897
Hence if the Mushaf that you have downloaded from that website is something for which the site owners have clearly put effort into programming it and preparing it in a form that they have invented, then they have the right to prevent you from publishing it unless that is done with their consent, and they have the right to protection of their copyright, not for the Book itself, but for the effort they put into programming.
But if the Mushaf is available on other websites and it is clear to you that they are not in fact the owners, and that the effort put into producing it did not come from them, and that its producers have given it to people for free, then you do not have to do respond to their request to remove it from your website.
If it so happens that you have erased the copy that you downloaded from their website, then you can upload another copy from those that are available on the Internet, which will mean that you have no need for that copy.
And Allah knows best.