It is permissible to recite the Qur'an in all conditions - walking, riding, or lying down.
`Abdullah ibn Mughaffal (may Allah be pleased with him) said:
I saw the Prophet (peace be upon him) reciting (Qur’an) while he was
riding his she-camel or camel. He was reciting the Surah of Al-Fat-h
or part of the Surah of Al- Fat-h very softly and in an attractive
vibrating tone. (Related by Al-Bukhari)
It is reported in Al-Bukhari and Muslim that `Aishah (may Allah be pleased with her) said:
The Messenger of Allah (peace be upon him) would recline in my lap when I was menstruating and recite the Qur'an.
It is reported that `Aishah (may Allah be pleased with her) used to recite her daily portion of Qur'an while she was reclining in bed.
Abu Musa Al-Ash`ary (may Allah be pleased with him) said:
"I recite the Qur'an in my Salah and in my bed."
Source: Etiquette of reciting the Qur'an