All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Please take it easy on yourself because you are not sinful for what happened, especially that you intended something good when you refrained from carrying the Quran from its place in the beginning. You avoided carrying it only because you were not in the state of Wudhoo’ (ritual purity).
Your action is in accordance with the statement of the majority of the scholars who are of the view that touching the Mus-haf (the hardcopy of the Quran) without being in the state of ritual purity is prohibited.
This issue is one about which the scholars differed in opinion. Some scholars are of the view that it is not a condition for one to be in the state of ritual purity in order to touch the Mus-haf. However, the view of the majority of the scholars is the predominant one.
Additionally, you are not sinful, Allah willing, by leaving the Mus-haf in the back seat if it was a pure spot and the Mus-haf was not subject to being humiliated or degraded. However, it would have been better if it was placed in a safe place where it would not be subject to such accidents, like placing it in the glove compartment or any other place where it would be more protected.
Allah knows best.