I quitted my night shift job after some months as a new grad: it was a difficult situation and I ended up physically sick. Should I mention the experience in my resume? (I don't have other experience and I have been post grad for 1 year. )

In the situation it’s best to be forthcoming with this information.
Explain to the future interviewers or Human Resource representative, and they will ask, circumstances surrounding your time in that specific position.
Ask any coworkers or managers that you worked with while in that position, if he/she would be a reference for yo.
***Make sure you ask prior to adding them to any future applications.
If you have anyone in that department that will be a reference for you, add them to your application.

Once about three years experience with good references is established, then omitting this time of employment history would be acceptable.