As the Qur'an was only revealed 15 years after their marriage and with the exception of the month of Ramadan no other month (AFAIK) was clearly quoted nor their marriage was clearly quoted in the Qur'an, but some interpretations may lead to a verse related to their marriage among them was:
And He found you poor and made [you] self-sufficient. (93:8)
Imam al-Qurtobi commented on it saying -see here in Arabic-:
أي فقيرا لا مال لك . فأغنى أي فأغناك بخديجة - رضي الله عنها
(And He found you poor ) Means: poor having no fortune, (and made you self-sufficient) meany Allah made you self-sufficient by Khadija may Allah be pleased with her.
Al-Baghawi -see here in Arabic- said the same after quoting a strange story on the authority of Sa'id ibn al-Musayyib quoting that of Muhammad () and his Uncle abu Talib joined the trade caravan of Maysarah the servant of Khadija which he used as an explanation of (93:7). Imam al-Fakhr ar-Razi even went further saying that Allah first made him self-sufficient by the education of abu Talib, then by the wealth of Khadija, then by that of abu BAkr, then by that of booty of war etc. -see here in Arabic-.
I couldn't find any source telling us about the month in which the prophet () married Khadijah (). The story of their marriage and how it was arranged is well elaborated in many different sources (however there might be some differences), but I couldn't find a quote of a date or month in which the marriage actually happened.
On the other hand I found a couple of ahadith and fatwas telling us that he married -at least some of- his other wives in the month of Shawwal:
the Prophet married Umm Salamah in Shawwal, and consummated the marriage with her in Shawwal. (Sunan ibn Majah, qualified as da'if)
'A'isha (Allah be pleased with her) reported:
Allah's Messenger (ﷺ) contracted marriage with me in Shawwal and took me to his house as a bride during Shawwal. And who among the wives of Allah's Messenger (ﷺ) was dearer to him than I, and' A'isha liked that the women (of her family) should enter the houses as brides during the month of Shawwal. (See for example in Sahih Muslim)
Imam an-Nawawi commented this hadith in his commentary of Sahih Muslim by the following words:
“This indicates that it is mustahabb to get married and to consummate the marriage in Shawwaal. Our companions [fellow-scholars] have stated that this is mustahabb, quoting this hadeeth as evidence for that.” (Source islamqa #12364)
- This fatwa islamweb #73028 /in Arabic) says that he married his second wife Sawda () in Shawwal too.
Note that Arabs used to avoid marrying in Shawwal due to their superstition:
The reason why the Arabs of the Jaahiliyyah were so superstitious about marrying during Shawwaal was that they believed that a woman would keep away from her husband like the she-camel that has lifted (shawwalat) its tail after being impregnated keeps away from the he-camel. (Source islamqa #12364)