As explained in The Muslim Brotherhood and the West: A History of Enmity and Engagement:
The first few months of the year  brought a new rush of anti-British rhetoric in the Egyptian press [reference 75 of chapter 4]. The Brotherhood was a leading purveyor of this invective, with the group's newspaper stressing the iniquitous role played by “the English,” not just in Egypt but also across the region (in Iraq, Palestine, Sudan, Jordan, Libya, India, Pakistan, and Iran).[reference 76]. A new take was provided on Britain's alleged historical animus against Islam by reference to nineteenth century prime minister William Gladstone, who was said to have told the House of Commons, while holding up a copy of the Qur'an, that there would be “no peace for the empire . . as long as this book exists.” [reference 77] This anecdote would become a familiar one in Brotherhood circles, offered as proof of Britain's abiding enmity toward Islam. [reference 78]
It is telling, for instance, that in 2015, the Brotherhood's spokesman in the United Kingdom felt it necessary to repeat a fanciful story that “demonstrated” nineteenth-century British prime minister William Gladstone's hatred for Islam. As described in Chapter 4, this tale had been circulating in Brotherhood circles since at least the 1950s, though Abdullah al-Haddad's version contained the added twist of Gladstone tearing up a copy of the Qur'an on the floor of the House of Commons.[reference 15 of Conclusion].
An early English-language example of the supposed quote, as published in Muslimnews International, May 1967, is:
Mr. Gladstone once said to the House of Commons (and he held a copy of Al-Qur'an in his hand) "So long as the Egyptians have got this book with them, we will never be able to enjoy quiet or peace in that land."
Note that the 1970 book (quoted in wikiquotes and another answer) Army officers in Arab politics and society, which is a translation of a 1966 Hebrew book, says:
In His memoirs Salah al-Din Sabbagh devoted a number of chapters to the principles in which he believed. ... He wrote :
"I don't believe in the democracy of the English, the Nazism of the Germans or the Bolshevism of the Russians. I am a Muslim Arab, and for me there is no substitute for this among all the views and philosophies; I want no comparison or preference among them as this is sterile and meaningless, for wherever I turn, I see the foreign wolf preying upon and torturing my nation— in the Mediterranean, Oman, the Persian Gulf, in the heart of the Arabian peninsula and near the tomb of the Prophet."
"There is no more murderous wolf for the Arabs and no deadlier foe of Islam than Britain. As for the Arabs they have been torn apart into small countries, communities and tribes that fight each other ... If Arabs seeking freedom rise up in Palestine, Egypt, Aden, the seven shaykhdoms and Iraq, the guillotine is sharpened for them and bombers are loaded with fire. Three hundred and fifty million Muslims are still groaning under the yoke of British imperialism. The bloody 'Lion-Heart' of the Crusaders' wars was an Englishman and so was Allenby, who conquered Jerusalem and said, 'Now the Crusades are over,' and so was Gladstone, who threw the Quran into a closet and said, 'There will be no quiet in the world as long as this remains,' and so was Cromer who said, 'Only this Quran impedes civilization.'"
A true quote by Gladstone about the Koran is, as published in his article Montenegro: A Sketch The Nineteenth Century, volume 1, issue III, May 1877, pages 360-379:
Before them, as before others, lay the trinoda necessitas, the alternatives of death, slavery, or the Koran. They were not to die, for they had a work to do. To the Koran or to slavery they preferred a life of cold, want, hardship, and perpetual peril.
I found a much older source of the claim, a German translation of a 1915 fatwa, published as Kriegsurkunden. 17. Fatwa des Scheich es-Saijid Hibet ed-Din esch-Schahrastani en-Nedschefi über die Freundschaft der Muslime mit den Deutschen Die Welt des Islams Bd. 4, H. 3/4 (Mar. 15, 1917), pp. 217-225, at page 220:
Ihr werdet Wohl alle von jenem Wort Gladstone's des englischen Ministers des Äußern, gehort haben das er im englischen Parlament gesagt hat:
"So lange noch der Koran des Mohammed als ein himmlisches Buch betrachtet wird, kann unsere Sache nicht vowärts gehen"
According to a recent book titled Nazis, Islamists, and the Making of the Modern Middle East, at page 42, this is a fabricated quote.