The modelling of emission spectra of molecules seen in interstellar clouds requires the knowledge of collisional rate coefficients. Among the commonly observed species, N$_2$H$^+$ is of particular interest since it was shown to be a good probe of the physical conditions of cold molecular clouds. Thus, we have calculated hyperfine-structure resolved excitation rate coefficients of N$_2$H$^+$(X$^1Sigma^+$) by H$_2(j=0)$, the most abundant collisional partner in the cold interstellar medium. The calculations are based on a new potential energy surface, obtained from highly correlated {it ab initio} calculations. State-to-state rate coefficients between the first hyperfine levels were calculated, for temperatures ranging from 5 K to 70 K. By comparison with previously published N$_2$H$^+$-He rate coefficients, we found significant differences which cannot be reproduced by a simple scaling relationship. As a first application, we also performed radiative transfer calculations, for physical conditions typical of cold molecular clouds. We found that the simulated line intensities significantly increase when using the new H$_2$ rate coefficients, by comparison with the predictions based on the He rate coefficients. In particular, we revisited the modelling of the N$_2$H$^+$ emission in the LDN 183 core, using the new collisional data, and found that all three of the density, gas kinetic temperature and N$_2$H$^+$ abundance had to be revised.
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