Well, yes there is, but it's slightly complicatd by the fact that the permutation action of $G$ on the (say right) cosets of $H$ introduces a sign. Also, you have to worry about "non-diagonal" blocks. Hence, if we let $T$ be a complete set of representatives for the right cosets of $H$ in $G,$ then ${\rm det} {\rm Ind}_{H}^{G}(\rho)[g] = \prod_{s,t \in T: sgt^{-1} \in H} {\rm det}(\rho(sgt^{-1}))$ if $\rho(1)$ is even, but is ${\rm sign}_{H}(g)\prod_{s,t \in T :sgt^{-1} \in H} {\rm det}(\rho(sgt^{-1})$ if $\rho(1)$ is odd, where ${\rm sign}_{H}(g)$ denotes the sign of the permutation of the right cosets of $H$ in $G$ induced by right multiplication by $g.$ In the 1970's, T. Yoshida did some work on "character-theoretic transfer", which exploited the determinant of the induced representation, especially when $\rho$ was linear. In some situations, Mackey type formulae for the determinant of the induced representation can simplify calculations.
Later edit: Perhaps a word about the use of Mackey type formulae. It can simplify things to look at the orbits of $\langle g \rangle$ on right cosets of $H$ in $G,$ in other words, to group contributions from the $(H, \langle g \rangle)$-double cosets. The factor ${\rm sign}_{H}(g)$, can be accounted for as above, suppose that thre are $k$ of these double costs, and that $t_1,t_2, \ldots, t_k$ are representatives, where the double coset of $t_i$ consists of $m_i$ right cosets of $H.$ Then the formula for the determinant of ${\rm Ind}_{H}^{G}(\rho)[g]$ may be expressed as ${\rm sign}_{H}(g)^{\rho(1)} \prod_{i=1}^{k} {\rm det}(\rho( t_{i}g^{m_i}t_{i}^{-1})).$