For a (molecular) graph, the first and second Zagreb indices ($M_1$ and $M_2$) are two well-known topological indices in chemical graph theory introduced in 1972 by Gutman and Trinajstić. Multiplicative versions of Zagreb indices, such as Narumi-Katayama index, multiplicative Zagreb index and multiplicative sum Zagreb index, have been much studied in the past. Let ${\bf{G}}(n,k)$ be the set of connected graphs of order $n$ and with chromatic number $k$. In this paper we show that, in ${\bf{G}}(n,k)$, Turán graph $T_n(k)$ has the maximal Narumi-Katayama index, the maximal multiplicative Zagreb index and the maximal multiplicative sum Zagreb index. And the extremal graphs from ${\bf{G}}(n,k)$ with $k=2$ or $3$ are determined with minimal values of these above indices.