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Let $G$ be $Sp(2n, \mathbb{R})$ or $SO^*(2n)$. We compute the Dirac index ofa large class of unitary representations considered by Vogan in Section 8 of[Vog84], which include all weakly fair $A_{\mathfrak{q}}(\lambda)$ modules and(weakly) unipotent representations of $G$ as two extreme cases. We conjecturethat these representations exhaust all unitary representations of $G$ withnonzero Dirac cohomology. In general, for certain irreducible unitary module ofan equal rank group, we clarify the link between the possible cancellations inits Dirac index, and the parities of its spin-lowest $K$-types.

Dirac index of some unitary representations of $Sp(2n, \mathbb{R})$ and $SO^*(2n)$