I was studying spherical basis as part of a physics course, and I stumbled the spherical representation of tensors. In Wikipedia (https://en.wikipedia.org/wiki/Spherical_basis) I found that any operator in the spherical basis satisfies the following commutation relationships:
$$[J_z,T_q]=\hbar q V_q$$
$$[J_{\pm},T_q]=\hbar \sqrt{(1\mp q)(2\pm q)}V_{q\pm 1}$$
However, I would like to know if it's possible to deduce this commutators or if these are just mere definitions, as all the textbooks and articles I've read just state these commutators and then work on other definitions built from these.
They are not definitions. If you have not come across raising and lowering operators, I would recommend to start on something simple. First you have \begin{equation} [J_x,J_y]= i\hbar J_z \end{equation} and cyclic permutations. You also have \begin{equation} [{\bf J}^2,{\bf J}]=0 \end{equation} Then you define \begin{equation} J_{\pm}=J_x\pm iJ_y \end{equation} and show a bunch of commutation relations, for instance \begin{equation} [\bf{J}^2,J_\pm]=0, \end{equation} and so on and express $\bf{J}^2$ in terms of $J_z$ and $J_\pm$. It is then possible to derive the common spectrum of ${\bf J}^2$ and $J_z$ (the eigenvalues of ${\bf J^2}$ can be written as $\hbar^2 j(j+1)$ and those of $J_z$ as $m\hbar$, with j either integer or half-integer and $m$ integer). All of this is material you will use over and over again in your studies. It is worth spending time on. I think you would be better off reading a standard textbook like Cohen Tannoudji, Quantum mechanics I, chapter VI, Angular momentum in quantum mechanics, see here
Finally, to get back to your question the spherical harmonics that you encountered are just a 3D representation of the angular momentum algebra above with integer values for $j$. The concept is much more general though and is also the basis for treating spin 1/2 particles.