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Motivated by recent nuclear magnetic resonance experiments, we calculated the spin susceptibility, Knight shift, and spin-lattice relaxation rate ($1/T_{1}T$) of the single-component molecular conductor [Ni(dmdt)$_2$] using the random phase approximation in a multi-orbital Hubbard model describing the Dirac nodal line electronic system in this compound. This Hubbard model is composed of three fragment orbitals and on-site repulsive interactions obtained using ab initio many-body perturbation theory calculations. We found fragment-orbital-dependent spin fluctuations with the momentum $\textbf{q}$=$\textbf{0}$ and an incommensurate value of the wavenumber $\textbf{q}$=$\textbf{Q}$ at which a diagonal element of the spin susceptibility is maximum. The $\textbf{q}$=$\textbf{0}$ and $\textbf{Q}$ responses become dominant at low and high temperatures, respectively, with the Fermi-pocket energy scale as the boundary. We show that $1/T_{1}T$ decreases with decreasing temperature but starts to increase at low temperature owing to the $\textbf{q}$=$\textbf{0}$ spin fluctuations, while the Knight shift keeps monotonically decreasing. These properties are due to the intra-molecular antiferromagnetic fluctuations caused by the characteristic wave functions of this Dirac nodal line system, which is described by an $n$-band ($n\geq 3$) model. We show that the fragment orbitals play important roles in the magnetic properties of [Ni(dmdt)$_2$].