I'm not sure if this is a bug or is intended, but the quest counts only the first upgrade of any artifact.
So, for example, if you upgrade 5 different artifacts once, you get +5 to the quest counter, but if you upgrade a single artifact 5 times, you only get +1.
If it's intended, maybe it'd be better to make it clearer in the quest description, something like "Upgrade 10 different artifacts once."