Does sitting idle for one month without work leads to depression?
Two answers:
The first, and most important one: do not ask any online forum whether you are getting depressed, since usually there will be many opinions about your situation from people who are not qualified in the field, be it psychiatry or psychotherapy. Please: if you do not feel well, and if your current situation is worrying you, seek help from your general physician or better if accessible to you a psychotherapist/ a psychiatrist with whom you can talk personally. There is no shame in seeking help from these persons, and in case you find out together with them that you truly need some kind of help, they will know ways of help that are based on evidence from empirical research and likely better and better fitting to your individual situation than suggestions on the internet.

Second: Your question's title implicitly asks whether sitting one month without work in front of a computer always leads to depression. It might and it might not depending on many factors (such as coping strategies with gratification crises, previous episodes of depressive disorders, social support, and many more, which differ from person to person). The danger of answers to a question stated like this is that yes-answers might lead you to think you are depressed, even though you are not, or no-answers lead you to think you aren't depressed even though you might need help. This is to say, the danger is to mistake such an answer for a diagnosis, which are two completely different things. A diagnosis needs to be done by a person qualified to do this, such as psychotherapists or psyhiatrists. They, too, will know further steps in case a diagnosis is given.