Je pensais qu'on part le travail a 19h; et qu'on quitte le travail (por toujours)

You could say "je pars du travail", meaning "I'm leaving from work". The situation is a bit muddled when using "quitter".

Je quitte mon travail. -- I'm quitting my job. Without context and in general, this means that you are actually terminating your job and no longer work there. To be even more explicit, you can say "je quitte mon emploi".

Je quitte le travail. -- I'm leaving work. This is usually understood as you are leaving from work and not as terminating your employment.

Quitter means 'to leave somewhere/someone' - and is used in such a manner, always with an object. 

 What is special about 'quitter' is that it also means ' to leave someone/somewhere for good/permanently/without intention of returning '. This particular meaning is exclusive to quitter (partir/sortir/laisser are not used in this context), but it is not the only situation in which quitter can be used.

Hi Anna,

Just to supplement what has already been written, the key to this issue is to understand the point being made at the very end of the lesson about "Grammar Jargon". Jargon maybe, but the issue of transitivity is vital to thorough understanding going forward.

If you were to get this issue clear in your mind it would help you a great deal IMHO.

Jim