I know that in context of time, if for example it is: 1:45 then I should say: "It is a quarter to two". But my question is about "half" in such context, for example, in case of 1:30. Should I say "It is a half past one."? I'm asking it because I didn't notice that people use this article in context of time.
As user2684291 points out in his comment, you shouldn't put any article there because it is a common expression of time. Same for the quarter, where the indefinite article is optional.