In most cases you would not capitalize either word on their own.
A possible exception might be if you were to use the word as a proper name, or as an abbreviated version of a proper name. For example, you would use capital letters to name the Organisation for Economic Co-operation and Development; after spelling it out in full the first time, you might shorten the name to "Organisation" on subsequent mentions: "The Organisation is expected to release its study next month." (In this particular case, however, you'd be better off calling it the "OECD," which is its preferred English-language acronym; the same is likely true for many groups with organization/organisation in their names, like OPEC, the WTO, etc.)
For government specifically, in some cases it may be proper to capitalize it as part of a proper name; for example, I believe that Her Majesty's Government (of the UK) and the Government of Canada both like to be referred to as such. Even in those cases, I would probably lowercase the word government on subsequent occurrences when using the word by itself, unless I were dealing with a house style guide that required otherwise.