Theoretically, yes it should be possible to derive the boiling point of diatomic nitrogen from fundamental forces. In fact, you don't even need to involve the strong force or weak force (or the strong nuclear force, which is sort of different). The strong forces bind the quarks together into nucleons and the nucleons together into nuclei, but they have essentially no effect on distance scales much larger than that of an atomic nucleus. So, for purposes of calculating the boiling point of nitrogen, you can treat the nucleus as basically a point charge. The only force that is relevant to calculating a boiling point is the electromagnetic force.
Now the bad news: even something as simple as calculating the energy levels of helium, with 2 electrons, is impossible to do analytically. To analyze the behavior of even just those two electrons (and He nucleus) in detail, you need to use either perturbation theory or a numerical simulation, or both. And of course, the complexity increases with the number of particles, so simulating the 14 electrons and 2 nuclei of a nitrogen molecule is absurdly complicated. Perhaps it's been done, but I'm not a condensed matter physicist so I wouldn't know where to look for a reference. Maybe someone else can provide you with that information.
If you were to calculate the boiling point of nitrogen, I believe the main effect that you'd take into account would be the instantaneous dipole interaction. According to the Wikipedia article, it gives an interaction energy in terms of the polarizabilities and ionization energies of the molecules. Those are the quantities that you would have to extract from your simulation and/or perturbative calculation of the dynamics of the nitrogen molecule, if you wanted to calculate the effect from first principles.
$$E_{AB}^{\rm disp} \approx -{3 \alpha^A \alpha^B I_A I_B\over 4(I_A + I_B)} R^{-6}$$
(that formula is actually for monatomic noble gases, it may not apply to diatomic molecules)
Once you get the interaction energy as a function of intermolecular separation $R$, you would then have to do either another numerical simulation, or a rather complicated calculation, to show that a large pool of nitrogen molecules subject to the given intermolecular force undergoes a phase transition at 77.36K (at standard pressure, I assume). There are various thermodynamic models you could use, some more accurate than others, but of course the more accuracy you want, the more computation power you'll need. I suspect that in order to get within a few degrees of the actual temperature, you would need to do something more computationally intensive than would be possible by hand.
Best Answer
I don't think it is possible to learn physics without math. Mathematics is the language of physics and you can't learn a subject without learning its langauage. (This is just my opinion though).
To start with particle physics, I think you should first learn quantum mechanics and special relativity. For quantum mechanics, grab a copy of Griffith or any other similar book. Knowledge of single variable calculus and differntial equation is needed though.For vector calculus, you can look into the first few chapters of Feynmann lectures vol 2. Along with these, you must also learn abstract Linear algebra (theory of vector spaces) , very basic group theory (definition of group and group actions) and multivariable calculus.
Once you are done with qm and special relativity, you will be ready for Quantum Field Theory.A nice book for QFT is Quantum Field Theory in nutshell by A. Zee. Also,, now you should learn about theory of group representations and lie groups. A good introductory book for this topic is Group and Symmetries by Yvette Kosmann-Schwarzbach. The last chapter deals with particle physics.
Note that this route won't make an expert on the topic but you will gain a good understanding of it.