Adding a 4 way switch to 2 existing 3 Way switches
This diagram describes what you want (source), it's just complicated by the box positioning. Normally you do it in a series


Three-way
Four-way
Three-way




I think you can tweak this to make it work. 

In Box #1, the incoming hot (the 14-2 black) should connect to the black screw of your switch. Wire the black from the 14-3 to your 14-2 neutral white.

Put your 4 way into Box #2 and wire it as shown in the diagram, but instead of nutting the black wires together, nut the white wire from your fixture (14-2) to one and the black to the other (it doesn't matter which you connect to).