Reduced transition probabilities have been measured for the neutron-rich $73,74,75\mathrm{Ni}$ nuclei with relativistic Coulomb excitation performed at the RIKEN Nishina Center. For the even-even Ni isotope the determined $B(E2;0^{+}\to 2^{+})$ value is compatible within errors with the result of a previous intermediate energy Coulomb excitation experiment, although being somewhat larger. For the odd Ni isotopes $B\left(E2\right)$ values have been determined for the first time. In the middle $\nu g_{9/2}$ shell nucleus $73\mathrm{Ni}$three peak candidates have been observed and tentatively assigned to the ($5/2^{+}$), ($13/2^{+}$), and ($11/2^{+}$) excitations. Two peak candidates have been observed in $75\mathrm{Ni}$ and tentatively assigned to the ($13/2^{+}$) and ($11/2^{+}$) levels. The measured $B\left(E2\right)$ values are in line with the large $B\left(E2\right)$ reported in $70\mathrm{Ni}$ and at variance with the small strength measured in $72\mathrm{Ni}$. Excitation energies compare well with the result of large-scale shell-model calculations, while $B\left(E2\right)$ values are larger than predicted. This could indicate an increasing contribution of proton excitations across the $Z=28$ shell closure when approaching $78\mathrm{Ni}$.