The NMR chemical shift arising from 4d electron angular momentum and 4d electron angular momentum and 4d electron spin dipolar-nuclear spin angular momentum interactions for a $4d^1$ system in a strong crystal field environment of trigonal symmetry, when the threefold axis is chosen to be the axis of quantization axis, has been examined. A general expression using the nonmultipole expansion method (exact method) is derived for the NMR chemical shift. From this expression all the multipolar terms are determined. We observe that along the (100), (010), (110), and (111) axes the NMR chemical shifts are positive while along the (001) axis, it is negative. We observe that the dipolar term (1/R3) is the dominant contribution to the NMR chemical shift except for along the (111) axis. A comparison of the multipolar terms with the exact values shows also that the multipolar results are exactly in agreement with the exact values around $R{geqslant}0.2$ nm. The temperature dependence analysis on the NMR chemical shifts may imply that along the (111) axis the contribution to the NMR chemical shift is dominantly pseudo contact interaction. Separation of the contributions of the Fermi and the pseudo contact interactions would correctly imply that the dipolar interaction is the dominant contribution to the NMR chemical shifts along the (100), (010), (001), and (110) axes, but along the (111) axis the Fermi contact interaction is incorrectly the dominant contribution to the NMR chemical shift.