Although this is not explicitly mentioned in any authentic hadeeth and, indeed, Allaah سبحانه و تعالى praises those who remember Him standing, sitting or while lying on their sides , many scholars believe that there is a relationship between the way one sits while reciting the Qur'aan and how much one benefits from said recital.
They suggest facing the Qiblah and sitting in the same posture that one sits in while in prayer. Perhaps sitting in a special manner to read the Qur'aan will remind the reader that what he is reading is special.
Many scholars, including Imaam an-Nawawi رحمه الله, record ahaadeeth stating that the best sitting is that wherein the person is facing the Qiblah. There is, for example, a hadeeth that states, "Everything has a most noble aspect to it and the most noble sitting is wherein one faces the Qiblah." 
If a person finds that sitting in a certain respectable fashion improves his reading and understanding of the Qur'aan, then it is good for him to sit in that fashion. However, it cannot be considered a must. 
(Soorah Aali 'Imraan, 3:191)