Ciao Francesco!
The answer to your question is yes, and the work of Bogomolov you are looking for about semistability of the tangent space (for minimal surfaces of general type indeed, no need of ampleness of the canonical bundle) is
"Holomorphic tensors and vector bundles on projective varieties", Math. USSR Izvestija 13/3 (1979) 499-555.
But let me give you a more general framework.
Given a Hermitian holomorphic vector bundle $(E,h)$ over a compact Kähler manifold $(X,\omega)$, if $(E,h)\to (X,\omega)$ is Hermite-Einstein then its dual as well as its symmetric powers (with the induced Hermitian structures) also are Hermite-Einstein.
Next, if $(E,h)\to (X,\omega)$ is Hermite-Einstein, then $E\to X$ is $[\omega]$-semistable (polystable, indeeed). This is the "easy" direction of the Kobayashi-Hitchin correspondence.
If a projective manifold $X$ has ample canonical bundle, then it admits a Kähler-Einstein metric $\omega$ of negative Einstein constant, i.e. a Kähler metric such that $\operatorname{Ric}(\omega)=-\omega$ (by the Aubin-Yau Theorem).
In particular $(T_X,\omega)\to (X,\omega)$ is Hermite-Einstein, and then so does any symmetric power of the cotangent bundle (with the induced Hermitian structure). This makes these vector bundles $[\omega]$-semistable, as we saw above.
Finally, in this case, begin $[\omega]$-semistable means that these vector bundle are $K_X$-semistable, since $[\omega]=[-\operatorname{Ric}(\omega)]=c_1(K_X)$.
You can see all this and much more on S. Kobayashi "Differential geometry of complex vector bundles".
P.S. For quite recent advances about semistability of the tangent sheaf of possibly singular varieties (even in the logarithmic setting), you might want to take a look to this paper by H. Guenancia.