Hi, my PC uses Intel Core i3-4130 (Haswell) 3.4 GHz Dual Core processor with integrated Intel HD Graphics 4400 (GT2) card, if I purchase and install a new graphics card, would I be able to play the game without getting a new processor like the Intel Core i7-4790 (Haswell) 3.6 GHz Quad Core?

Yes you will be able to play the game. Then it depends from your GPU.

To give you a tip.

A GTX 980TI with an i3-4130 on 1080p (medium settings) comes to 40-60 fps (low to high)

Of course you don't wanna buy a GTX 980TI 6GB Hybrid with that processor.

What that means. You will be fine as long you don't wanna play at 1080p.

Graph for proof: http://media.gamersnexus.net/images/media/2015/game-bench/fallout-4-cpu-benchmark-1080-m.png