There are a few things going on here. Hebrew verbs have paradigms, e.g. different conjugations, and these paradigms determine different meanings in a way similar to, but not really the same as, how in indo-european languages you conjugate differently based on direct versus indirect object.
In this case, quaran means "to shine" in it's basic Qal paradigm (the most common, default paradigm) but in the Hiphil paradigm it means "to have horns". It's the same lemma, but based on the conjugation we can determine the meaning. This is an extreme example of meaning changing based on paradigm.
Now there must be some relationship between horns and shining to explain why the Hiphil is to have horns, and that is that horns in cattle cultures are a symbol of power and strength. Do a word search in the old testament on "horn" and it's very fascinating. So with this knowledge, it begins to make more sense. Hiphil is often (but not always) interpreted as a causative form. So if in Qal the verb is "to cook", then the Hiphil might be "to cause someone to cook". Again, this is just one useful way of thinking about Hiphil but these paradigms defy simple categorization. But at least here it makes sense that a causative agent for "shining" would be "power" and thus "horn".
As a fun-fact bonus, it said that a mistranslation of Qaran caused Michelangelo to paint Moses as having horns, when in reality his face was shining. But in retrospect, I wonder if Michelangelo knew exactly what he was doing, as portraying Moses as a source of power by decorating him with horns would be a clever and historically appropriate illustrative device. Then again this could be my own personal whimsy.
For more information, see the Theological Wordbook of the Old Testament entry on Qaran, from which the above information about Qaran, Hiphil, and Michelangelo was sourced:
Coppes, L. J. (1999). 2072 קָרַן. R. L. Harris, G. L. Archer Jr., & B. K. Waltke (Eds.), Theological Wordbook of the Old Testament (electronic ed., pp. 815–816). Chicago: Moody Press.
Quaran versus 'wr
'wr is a verb that means "to give light". See gen 1.15. Note that it is not the noun 'or which means "light". 'or is never translated as "shine" but the verb 'wr is translated as shine. One is a verb, the other a noun.
Although there is certainly semantic overlap between 'wr and quaran, the former focuses on "giving light" whereas the latter suggests light emitting from a source of power. Both of these semantic regions overlap in what we would call "to shine", but they do have different semantic ranges so are not exact synonyms.
Explaining the author's choices
What's the reason behind using this term here?
Because the author is emphasizing the glorification/radiance of Moses due to the power within him, rather than making a statement about illumination.
The Israelites feared Moses because of his radiance - the hebrew word is literally "dread" (Ex 34.30).
In Psalm 104.2 God is described as "clothed" with light.
you who cover yourself with light as with a garment, who stretch out the heavens like a tent curtain,
So now Moses spends 40 days and nights in the presence of God, and his face is shining with light, but that light is the clothing of God's power; it is the presence of God that is the source of that light, and this power is what the Israelites feared.
If it was just a light source -- something shining -- then this does not cause us to dread or be terrified.
Therefore the use of Quaran reinforces this notion of power better than the use of 'wr.
Could it be related with the golden calf?
Unlikely. If you are suggesting some kind of wordplay, which is very common in the old testament, you should provide more examples.