No. The power of begetting signifies principally the divine essence and not the relation only (nor does it signify the essence as identified with the relation, so as to signify both equally) because that by which the Father begets is the divine nature, in which the Son is like to Him.
The power of begetting signifies the divine nature directly, but the relation indirectly.
For although paternity is signified as the form of the Father, nevertheless it is a personal property, being in respect to the person of the Father, what the individual form is to the individual creature.
Now the individual form in things created constitutes the person begetting, but is not that by which the begetter begets, otherwise Socrates would beget Socrates. So neither can paternity be understood as that by which the Father begets, but as constituting the person of the Father, otherwise the Father would beget the Father.