$\begin{align}  & \text{First convert to decimal system} \\ & {{2232}_{4}}=2\times {{4}^{3}}+2\times {{4}^{2}}+3\times {{4}^{1}}+2\times {{4}^{0}} \\ & =128+32+12+2 \\ & ={{174}_{10}} \\ & \text{Convert 17}{{\text{4}}_{\text{1}0}}\text{to base 6} \\ & \text{gives 17}{{\text{4}}_{10}}={{450}_{6}} \\\end{align}$