Yes, it us permissible to shred papers with Quranic verses but don't put the shred papers in a bad place. You can recycle such papers or bury or put it in the water of sea, river or will.
Wassalam.
Sayyed Mohammad al-Musawi is originally from Iraq and heads up the World Ahlul Bayt Islamic League in London. Other than being involved in various humanitarian projects, he frequently responds to religious questions. In the past, he has also spent significant time in India guiding the community.