There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]_q=AB-qBA$, one of the most general ones (though i do not now if this is "symmetric" enough for your purposes) is the following one: $$ \big[A,[B,C]_{q_1}\big]_{q_2}+q_2\big[B,[C,A]_{q_1}\big]_{q_2^{-1}}+\big[C,[A,B]_{q_1q_2}\big]=0 $$ which is valid for arbitrary values of the parameters $q_1$, $q_2$.
Notice that both identities stated by the OP are compatible with it:
- The first one can be recovered for: $q_1=q$, $q_2=1$,
- while a particular case (for $k=-m$) of the second one for: $q_1=q^n$, $q_2=q^m$.
Edit: I was thinking that the second identity of the OP might be combined with the 2-parameter deformation given here to construct a further 3-parameter deformation of the Jacobi identity. But i am not sure on its exact form, i 'll try to think a little on it and come back if something interesting comes up.