The colour of transition metal compounds arises due to d–d electronic transitions within partially filled d-orbitals. When light falls on such a compound, electrons in lower energy d-orbitals absorb specific wavelengths and get excited to higher energy d-orbitals, producing characteristic colours.
Now, let us analyze the given ions:
1. \( \text{(A) Cr}^{3+} \): Electronic configuration = [Ar] 3d$^{3}$. Since the d-orbitals are partially filled, d–d transitions are possible, so compounds of Cr$^{3+}$ are coloured.
2. \( \text{(B) Co}^{2+} \): Electronic configuration = [Ar] 3d$^{7}$. With partially filled d-orbitals, d–d transitions occur, hence Co$^{2+}$ compounds are coloured.
3. \( \text{(C) Ni}^{2+} \): Electronic configuration = [Ar] 3d$^{8}$. This ion also has partially filled d-orbitals, allowing d–d transitions, making Ni$^{2+}$ compounds coloured.
4. \( \text{(D) Zn}^{2+} \): Electronic configuration = [Ar] 3d$^{10}$. Here, the d-orbitals are completely filled. Since no d–d transitions are possible, compounds of Zn$^{2+}$ are colourless.
Therefore, the correct answer is option (D) Zn$^{2+}$.