Is there a Quranic verse or a specific supplication that can be used to expedite a girl's marriage?

There is no specific

ma'thur (tradition-based) supplication for this matter. However, comprehensive supplications for the good of this world and the Hereafter include it, such as: "Our Lord, grant us good in this world and good in the Hereafter and protect us from the torment of the Fire," and "O Allah, set right for me my religion which is the safeguard of my affairs..." A Muslim may supplicate with whatever he wishes, as long as it is not a sin or cutting off ties of kinship, for Allah is near and responsive to whoever supplicates to Him, as stated in His words, "And when My servants ask you concerning Me, then [answer them], I am indeed near. I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me [by obedience] and believe in Me that they may be [rightly] guided" and "And your Lord said, 'Call upon Me; I will respond to you.'"