Will nitrogen molecules form strong induced dipole-induced dipole bonds with one another? Why or why not?

”Strong” is a relative term. The induced dipole-induced dipole forces between nitrogen molecules are weaker than the attractions between permanent dipoles or hydrogen bonding interactions. Comparing only with other species whose strongest intermolecular attractions are induced dipole-induced dipole, we still find that the attractions in nitrogen are relatively weak.
Strength of induced dipole-induced dipole attractions increases as the number of electrons increases. Number of electrons increases as molecular weight increases, whether we move to more massive atoms in the formula or to a formula containing more atoms, so we often look at molecular weight to predict the strength of induced dipole-induced dipole interactions. The halogens provide an instructive example. All of them form diatomic molecules, like nitrogen. The two lightest, fluorine and chlorine, are gases at room temperature. Bromine, with a molar mass close to 160 g, is a liquid at room temperature, while iodine, with a molar mass of about 254 g, is solid at room temperature.
The molar mass of molecular nitrogen is 28 g. A few gases, including hydrogen, helium, methane, and neon, have lower molar masses, while those of diatomic oxygen and fluorine and of argon are a little more. Most molecules that feature only induced dipole-induced dipole interactions are more massive, so on this basis we can state that the induced dipole-induced dipole attractions are not very strong.