Yes we do. We just don't have a constitutional document that you can point at and say "That's the constitution" like the US does.
The Government, or to give it its current full title, Her Majesty's Government of the United Kingdom of Great Britain and Northern Ireland (there's a clue there—the Government belongs to Her Majesty, so she is not a part of it), consists of the ministers, not the monarch, who acts on the advice of the ministers/government.. The word government is also sometimes used casually to refer to all of Parliament, which again sits under the monarch.
I imagine this is historically related to the emergence of Parliament and the Government from the gradual weaning of power away from the monarch, rather than the from-scratch construction of a system of government. If it makes you feel better, think of government in this context as a homonym (or a polyseme if you prefer) with the word that is used elsewhere.