Is there a supplication or a Qur'anic verse to prevent a girl from rejecting a marriage proposal, and is Istikhara prayer related to this, given the rejection by girls to whom proposals have been made?

There is no specific supplication for facilitating marriage to a particular woman. However, supplication in general is beneficial. Istikhara (seeking guidance from Allah) is prescribed when contemplating a permissible matter, so that the servant may entrust his Lord with choosing what is best for him.