When traveling can I hold a Qur'an even when I am not praying because of the way luggage is handled in planes, I want to carry it with me.

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

If a person is in minor or major impurity or if a woman is in her period then touching or carrying the Qur'an directly or indirectly is forbidden. It is also forbidden to touch its footnotes and cover according to the preponderant opinion of Muslim scholars.
However, al-Ahnaf and al-Hanabilah permitted carrying the Qur'an with its cover, while al-Malikiyyah and al-Shafiyyah do not agree with them.
Therefore, it is permissible for the traveler who cannot make ablution to carry the Qur'an with him. This opinion is very reasonable since they believe that the prohibition of touching the Qur'an during the state of impurity applies only to direct touching, while touching the cover does not touch what is inside.
Allah knows best.