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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 $q_1$, $q_2$.