The Dyson series is an expansion in the coupling constant ##g##. Obviously you are evaluating the quoted 2nd-order loop corrections to the quark and gluon self-energies as well as the gluon 3-vertex to get the counter terms to evaluate ##Z_1##, ##Z_2##, and ##Z_3##, from which ##Z_g## follows through a Slavnov-Taylor identity (which maybe is Eq. (75), which you didn't quote). Of course since you have the said ##Z##-factors only to order ##g^2## (or order ##\alpha_s=g^2/(4 \pi)##), you can determine only the counterterm contributing to ##Z_g## to this order, and thus you have to expand the expression for it also up to order ##\alpha_s##.
In the here obviously applied minimal-subtraction scheme you need to find the coefficients to ##1/\epsilon## order by order perturbation theory. This determines the counter terms order by order. At higher loop order you have to take care of the subdivergences by using the corresponding counter terms of subdiagrams.