Reiki is a Japanese form of alternative medicine called energy healing. Reiki practitioners use a technique called palm healing or hands-on healing through which a "universal energy" is said to be transferred through the palms of the practitioner to the patient in order to encourage emotional or physical healing.
According to some scholars in Iran, reiki is permissible.
In order to do reiki properly according to conventional methods, one does not recite anything except for a few words (which are optional and related to the technique), so reciting Qur'an here is not a factor. If one is relying on recitation of Qur'an or something else as the primary method, it is no longer reiki.
Of course there is no harm in reciting Qur'an or playing it in the background, and there is no harm in reciting Qur'an in the hope of or with the intent of healing.