The answer of this question is yes the Qur'an describes the moon as a corpus that is lit, but no it doesn't explicitly say it is lit by reflected sunlight.
There are a couple of verses that describe the "light" or kind of lightening or illumination of the sun and the moon.
It is He who made the sun a shining light and the moon a derived light and determined for it phases - ... (10:5)
And made the moon therein a [reflected] light and made the sun a burning lamp? (71:16)
and a rather indirectly in:
And constructed above you seven strong [heavens] (12) And made [therein] a burning lamp (78:12-13)
In these verses the attribute or example of the moon was that of a النور (an-Nur) which refers in Arabic to anything that is opposed to darkness (ضدّ الظُّلمة ) and anything that makes things visible (Source: معجم المعاني الجامع - ما يُبَيِّن الأشْياءَ ويُرِي الأبْصَارَ حقيقتها ).
While the sun has been named الضياء (a-Dyaa' coming from ad-Daw'u الضوء which is a synonym of an-Nur at least when we don't want to be too specific ) and السراج (as-Siraj) which actually is an oil lamp (that gives light). Note that the term ad-Diya' refers toa source of light or something that gives light!
Here a statement from the dictionary قاموس المعجم الوسيط
الضَّوْءُ : النُّورُ .
وهما مُتَرَادِفانِ ، أَو الضوءُ أَقوى وأَسطعُ من النورِ ، أَو الضوء لما بالذاتِ كضوءِ الشمس والنارِ ، والنورُ لما بالعَرَضِ والاكتساب من جسم آخر
Ad-Daw'u: an-Nur and they are synonyms or ad-Daw'(u) is stronger and brighter than an-Nur or ad-Daw'(u) refers to anything that gives a light byitself like the sun and the fire while an-Nur appears as giving light or reflects it from another source (corpus).
The Qur'an therefore makes a distinction between sun and moon saying the sun is a source of light while the moon is a corpus that appears as giving light or reflects light, but it does not say that the reflected light actually is sunlight.