Do I need to install any specific toolbox in addition to SPM, Fieldtrip?
No, you don't have to install anything else.
How do I go about reproducing the resting state analysis results published in this paper?
By running the script tutorial_omega.m:
brainstorm3/toolbox/script/tutorial_omega.m at master · brainstorm-tools/brainstorm3 · GitHub
Or by doing it manually, as instructed in the online tutorial:
https://neuroimage.usc.edu/brainstorm/Tutorials/RestingOmega
Maybe something changed in Brainstorm since it was published, or something is set up in a wrong way on your computer.
To get the message that caused the error: check the execution report after it crashes. Copy-paste the error messages here if you don't understand them.
(menu File > Report viewer, or bst_report('Open','Current'))