All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
We have previously underlined the religious ruling on making a promise, its fulfillment, and effects if there was no oath associated with the promise in Fatwa 84051.
As for making a promise on the Quran, we do not understand what you mean exactly. If you refer to swearing by the Quran to fulfill a promise, then this is considered a binding oath that should be fulfilled. In case you failed to fulfill such an oath, it is incumbent on you to pay expiation for breaking an oath. Ibn Muflih wrote in his book Al-Mubdi‘, "If a person swears by the words of Allaah or by the Quran, then it is considered a binding oath according to the majority of Muslim scholars." For more benefit, please refer to Fatwa 81501.
On the other hand, if you mean placing one's hand on a copy of the Quran for the purpose of emphasizing one's promise, this is not considered a binding oath; rather, it is to emphasize the promise.
Allaah Knows best.