Is it permissible to swear on the Qur'an to someone who is not a forbidden relative (mahram) to me, such as a girlfriend or fiancée, and is it forbidden because the relationship itself is forbidden?

It is obligatory for a Muslim to fear Allah Almighty and to avoid forbidden relationships between men and women, for they are a cause of corruption of hearts and Allah's wrath, and they are among the steps of Satan. Whoever falls into any of that must offer sincere repentance and abandon the sin. As for swearing on the Mus'haf (Quran), it is a binding oath that requires an expiation for breaking an oath (kaffarat yamin) upon defaulting. The ruling on the oath varies according to what is sworn upon; fulfilling it may be obligatory or recommended, or breaking it may be obligatory if what was sworn upon was to abandon an obligation or commit a forbidden act.