  We report $^{75}$As NQR measurements on single crystalline
Ca(Fe$_{1-x}$Co$_x$)$_2$As$_2$ ($0\leq x \leq 0.09$). The nuclear spin-lattice
relaxation rate $T_1^{-1}$ as a function of temperature $T$ and Co dopant
concentration $x$ reveals a normal-state pseudogap-like phase below a crossover
temperature $T^*$ in the under- and optimally-doped region. The resulting
$x$-$T$ phase diagram shows that, after suppression of the spin-density-wave
order, $T^*$ intersects $T_c$ falling to zero rapidly near the optimal doping
regime. Possible origins of the pseudogap behavior are discussed.
