the coupling constant in QFT is scale dependent. This scale \mu, it seems to me, is a mathematical tool used to regularize the infinities that arise at the loop level. Thus physical quantities have to be independent of this scale \mu. This implies that the coupling 'constant' is a function of \mu.
I read an article by Wilczek (http://www.frankwilczek.com/Wilczek_Easy_Pieces/298_QCD_Made_Simple.pdf) and it contains the following senctence:

I don't quite understand the justification why one can associate this mathematical, unphysical scale \mu with the very physical momentum of the gluon.
Does this mean that the coupling can have different values within a Feynman graph depending how much momentum is carried by its attached gluon?

The scale is a typical heaviest mass or invariant energy carried by the system. It is not much better defined than an order of magnitude, however the dependences are at worse logarithmical so this is not a significant problem.