Analyze Statement I

Both trimethylamine, $\mathrm{N(CH_3)_3}$, and trimethylphosphine, $\mathrm{P(CH_3)_3}$, possess a lone pair on the central atom.

• $\mathrm{N(CH_3)_3}$ can donate the lone pair on nitrogen to a metal center.
• $\mathrm{P(CH_3)_3}$ can donate the lone pair on phosphorus to a metal center.

Therefore, both can act as Lewis base ligands and form transition metal complexes.

So, Statement I is correct.

Analyze Statement II

Statement II says that since N and P belong to the same group, the nature of bonding of $\mathrm{N(CH_3)_3}$ and $\mathrm{P(CH_3)_3}$ with transition metals is always the same.

This is incorrect.

Reason:

Although N and P are in the same group, their bonding behavior with transition metals differs significantly.

• Amines like $\mathrm{N(CH_3)_3}$ are primarily $\sigma$-donor ligands.
• Phosphines like $\mathrm{P(CH_3)_3}$ are $\sigma$-donors and can also participate in $\pi$-acceptor/back-bonding to some extent.

Thus, the metal-ligand bonding characteristics are not always the same.

Hence, Statement II is incorrect.

Conclusion

• Statement I: Correct
• Statement II: Incorrect

Therefore, the correct option is:

$\boxed{\text{D}}$

Comparison with stored answer

Stored correct answer: $\mathrm{D}$

Our derived answer is also $\mathrm{D}$, so they agree.