There is not, as far as we know, any du`a specifically for finding a good spouse that has been reported from the Prophet (peace and blessings of Allah be upon him), but you can pray to Allah to bless you with a righteous spouse.
The Prophet (peace and blessings of Allah be upon him) taught us Du`a al-Istikharah , which means asking Allah to choose for us what is best for our religious and worldly interests in all matters, religious and worldly.
This du`a was narrated by Jabir ibn ‘Abd-Allah (may Allah be pleased with him) who said: the Messenger of Allah (peace and blessings of Allah be upon him) used to teach us to seek guidance (istikharah) in all our affairs, just as he would teach us a Surah from the Quran. He said: Whenever any one of you is concerned about a matter, let him pray two rak’ahs of non-obligatory prayer, then let him say: Allahumma inni astakhiruka bi ‘ilmik, wa astaqdiruka bi qudratik, wa asaluka min fadlik al-azim. Fa innaka taqdiru wa la aqdir, wa ta’lamu wa la a’lam, wa anta ‘allamul-ghuyub. Allahumma in kunta ta’lam anna [hadhal-amr] khayrun li fi dini wa ma’ashi wa ‘aqibati amri, faqdurhu li, wa yassirhu li, thumma barik li fih. Wa in kunta ta’lam anna [hadhal-amr] sharrun li fi dini wa ma’ashi wa ‘aqibati amri, fasrifhu ‘anni wasrifni ‘anhu waqdur li al-khayr haythu kan. Thumma ardini bih (O Allah, I seek Your guidance (in making a choice) by virtue of Your knowledge, and I seek ability by virtue of Your power, and I ask You of Your great bounty. You have power, I have none. And You know, I know not. You are the Knower of hidden things. O Allah! If in Your knowledge, (this matter [ and he should mention it here]) is good for my religion, my livelihood and my affairs, immediate and in the distant future, ordain it for me, make it easy for me and bless it for me. And if in Your knowledge (this matter [ and he should mention it here]) is bad for my religion, my livelihood and my affairs, immediate and in the distant future, then turn it away from me, and turn me away from it, and ordain for me the good wherever it may be, and make me pleased with it). (Narrated by al-Bukhari, 1109)
Qualities to look for in a husband
There is nothing which it is more important for both parties to focus on than religious commitment and attitude.
With regard to the husband, the Prophet (peace and blessings of Allah be upon him) advised the guardians (walis) of women:
“If there comes to you one whose attitude and commitment to religion pleases you, then marry him (to your female relative), for if you do not do that, there will be much trouble and corruption in the land.” (Narrated by al-Tirmidhi, 1084; Ibn Majah, 1967)
Qualities to look for in a wife
With regard to the wife, the Prophet (peace and blessings of Allah be upon him) gave the following advice:
“Women are married for four things : wealth, lineage, beauty and religious commitment. Marry the one who is religious, may your hands be rubbed with dust! (i.e., may you prosper).” (Narrated by al-Bukhari, 4802; Muslim, 1466).
Al-Hafiz ibn Hajar said: “The meaning is that what is appropriate for the man who is religious and chivalrous is that religion must be the ultimate focus in all things, and especially with regard to the choice of a lifelong companion. So the Prophet (peace and blessings of Allah be upon him) commanded him to seek a religious woman , since religion is his ultimate goal.” (Fath al-Bari, 9/135)
How can the Quran be your source of guidance?
With regard to the second part of your question, about letting the Quran be your guidance :
If you mean using it for divination, like astrology, this is haram. One of the innovated ways in which people do this is to open the Mus-haf at random, and if their gaze falls on a verse which speaks of mercy or reward, they think this means that what they want to do is good! And if they see a verse which speaks of punishment, they think that what they want to do is bad. There is no basis for this in the religion; this is one of the ways in which the shayatin (devils) misguide the believing slaves of Allah.
But if what you mean by letting the Quran be your guidance is following it and acting in accordance with its rulings, this is the way to achieve true happiness, and there is no happiness apart from this, which is adherence to the Quran and Sunnah. This is the only way for mankind to be guided and saved by Allah on the Day of Resurrection. Concerning this Allah says (interpretation of the meanings):
“This is the Book (the Quran), whereof there is no doubt, a guidance to those who are Al-Muttaqun [the pious].” [al-Baqarah 2:2]
“Verily, this Quran guides to that which is most just and right.” [al-Isra 17:9]