  We propose a new ion-trap geometry to carry out accurate measurements of the
quadrupole shifts in the $^{171}$Yb-ion. This trap will produce nearly ideal
harmonic potential where the quadrupole shifts due to the anharmonic components
can be reduced by four orders of magnitude. This will be useful to reduce the
uncertainties in the clock frequency measurements of the $6s~{^2}S_{1/2}
\rightarrow 4f^{13} 6s^2 ~{^2}F_{7/2}$ and $6s~{^2}S_{1/2} \rightarrow 5d
~{^2}D_{3/2}$ transitions, from which we can deduce precise values of the
quadrupole moments ($\Theta$s) of the $4f^{13} 6s^2 ~{^2}F_{7/2}$ and $5d
~{^2}D_{3/2}$ states. Moreover, it may be able to affirm validity of the
measured $\Theta$ value of the $4f^{13} 6s^2 ~{^2}F_{7/2}$ state where three
independent theoretical studies defer almost by one order in magnitude from the
measurement. We also perform calculations of $\Theta$s using the relativistic
coupled-cluster (RCC) method. We use these $\Theta$ values to estimate
quadrupole shift that can be measured in our proposed ion trap experiment.
