The collision processes between N³⁺ ions and He atom is of great significance in astrophysics, interstellar space and laboratory plasma environment. The single and double charge transfer processes for the collisions of N³⁺ with He atom are studied by using the quantum-mechanical molecular-orbital close-coupling (QMOCC) method. The ab initio multireference single- and double-excitation configuration interaction (MRD-CI) method was employed to obtain the adiabatic potentials and the radial and rotational coupling matrix elements that are required in the QMOCC calculation. In the present QMOCC calculations, 10 ¹Σ states, 8 ¹Π states and 4 ¹△ states are considered, and total single and double charge transfer cross sections and state selection cross sections are calculated in the energy region from 3.16×10^-3 eV-24 keV (i.e., 2.25×10^-4 eV/u - 1.73 keV/u). Compared our results with the previous theoretical and experimental results, it can be found that our results agree well with the experimental values for the total double charge transfer (DCT) cross sections. For the total single charge transfer (SCT) cross sections, our QMOCC results are slightly higher than the experimental results in the energy region 0.2-11 eV/u. When the energy higher than 11 eV/u, the present QMOCC results are in good agreement with the experimental results. The total SCT cross section is significantly larger than the total DCT cross section, so SCT processes is the dominant reaction process. For the SCT process, it can be observed that the charge transfer to N²⁺(2s²p^{2 2}D) and N²⁺(2s²2p ²P^o) is very important. It should be noted that although we and Liu et al. [Phys. Rev. A 84, 042706, (2011)] both used the QMOCC method to study the charge transfer cross section, our calculation results are still significantly different from their calculation results. It is due to that Liu et al. calculations only considered 10 ¹Σ states and 8 ¹Π states, and ignored the effect of ¹△ states.