[Vuejs]-Error: Can't resolve 'child_process' and Error: Can't resolve 'fs'

AFAIK google-spreadsheets is a node package that depends on node specific libraries like fs

Inferring from the error message you provided and the fact you are trying to use this package with Vue leads me to believe you are trying to use google-spreadsheets in the browser directly.

I’m not entirely sure what your use case is but, if you are set on using this package I would suggest making a separate node project to play with and potentially use that node project to send data to your vue project.