Due diligence. The article specifies the condition clearly. A minimum competence review provides lots of explicit examples.
The broken generators in your case are the $N^2-1$ generators not in $su(N)_V$, the axial ones, which do not close to a group.
You must first compute the infinitesimal transforms of your $N^2-1$ operators $\bar \psi \gamma^5 T^a\psi$ under these $N^2-1$ axial transforms, and take their QCD vacuum expectation values.
If such a value does not vanish, i.e. if the operator transforms inhomogeneously, the operator interpolates a Goldstone mode. Such non-vanishing v.e.v.s are nontrivial null eigenvectors of the Goldstone mass matrix.
In your case, all those $N^2-1$ do. In fact, the pions and the SSB charges (now gone), have the same quantum numbers, in a 1-to-1 correspondence.
Example Let us illustrate this for N=2, which is to say
$SU(2)_L\times SU(2)_R\to SU(2)_V$ easier to picture in the algebraic equivalent language $SO(3)\times SO(3) \to SO(3)_V$. So you know, from your σ model (you wouldn't be asking this question unless you mastered this!), that the SO(4) vector $$ \begin{pmatrix}\pi^1\\ \pi^2\\ \pi^3\\ \sigma \end{pmatrix}$$ has its components scrambled pairwise by 6 generators. The SO(3) vector ones rotate the three pion components among themselves, and the 3 axials (not closing to a Lie algebra, of course) scramble each pion with the scalar σ.
The 0th component of the three vector currents is
$$ \vec V^0= \vec \pi \times \partial^0 \vec \pi,
$$
which you'll space-integrate to the charge generating the three isospin rotations;
while the same for the three axial currents is
$$
\vec A^0 = -\partial^0 \sigma ~\vec \pi+ \sigma \partial^0 \vec \pi .
$$
Consequently,
$$
\delta_A \vec \pi = \vec \theta_A \sigma ,\\
\delta_A \sigma = -\vec \theta_A \cdot \vec \pi,
$$
where I have been cavalier with normalization constants.
Now for the crucial part: For vacuum values $\langle \vec \pi\rangle=0$ and $\langle \sigma\rangle=v$, it follows that
$\langle \delta \sigma \rangle =0$, and, crucially,
$$
\langle \delta \vec \pi \rangle =\vec \theta_A v\neq 0,
$$
so each component is a Goldstone boson, a null eigenvector of the Goldstone mass matrix. You also see the direct connection between the three axial parameters with the three pions, i.e. the three SSBroken axial charges.
