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We consider a class of unbounded nonhyperbolic complete Reinhardt domains $$D_{n,m,k}^{\mu ,p,s}:=\left\{(z,w_1,\cdots ,w_m)\in {ℂ}^n\times {ℂ}^{k_1}\times \cdots \times {ℂ}^{k_m}:\frac{\parallel w_1{\parallel }^{2p_1}}{{\mathrm{e}}^{-{\mu }_1{\parallel z\parallel }^s}}+\cdots +\frac{\parallel w_m{\parallel }^{2p_m}}{{\mathrm{e}}^{-{\mu }_m{\parallel z\parallel }^s}}<1\right\},$$ where $s$, $p_1,\cdots ,p_m$, ${\mu }_1,\cdots ,{\mu }_m$ are positive real numbers and $n$, $k_1,\cdots ,k_m$ are positive integers. We show that if a Hankel operator with anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $A^2\left(D_{n,m,k}^{\mu ,p,s}\right)$, then it must be zero. This gives an example of high dimensional unbounded complete Reinhardt domain that does not admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.