The NMR chemical shift arising from 4d electron angular momentum and 4d electron spin dipolar-nuclear spin angular momentum interaction for a $4d^1$ system in a strong crystal field of octahedral symmetry, when the threefold axis is chosen as the quantization axis, has been investigated. A general expression using a nonmultipole expansion method is derived for the NMR chemical shift. From this expression all the multipolar terms are determined. We find that the nonmultipolar results for the NMR chemical shift ${Delta}B$, is exactly in agreement with the multipolar results when $R {ge} 0.20$ nm. It is also found that the 1/$R^7$ term contributes to the NMR chemical shift almost the same as the 1/$R^5$ in magnitude. The temperature dependence analysis of ${Delta}B$/B(ppm) at various values of R shows that the 1/$T^2$ term has the dominant contribution to the NMR chemical shift but the contributions of other two terms are certainly significant for a $4d^1$ system in a strong crystal field of octahedral symmetry when the threefold axis is chosen to be the axis of quantization.