Often, yes, but this is highly dependent on the task -- and sometimes the sub-task. For example, when writing finding the correct word for an idea is motivating, but proofreading for spelling is a chore. I can name a good number of tasks that I do because I am internally motivated, including looking up definitions, washing my hair, organizing my bookshelves, making checklists and flowcharts, grocery shopping, gardening... but just as many (and probably more) that I don't do unless I really feel I need to because it is expected of me.

I think that part of this is that unmotivating tasks tend to stick in your mind as always present and demanding, but the things you enjoy doing are all too fleeting. It is also possible that you are unconsciously defining conscientiousness as "things you should do but don't want to", in which case by definition they will require an external force.