As Ben Jamin says, "Den hellige ånd" is correct according to Norwegian rules for capitalization. This is also what Språkrådet recommends, see:
Stor eller liten forbokstav
However, you will often see "Den Hellige Ånd". I don't know if this capitalization is meant to express deference, or if it is influence from English capitalization - it might be both. Not even the Church of Norway seems to be consistent. On its website, it starts with "Den hellige ånd", then comes "Den Hellige Ånd" (in a direct quote from Luther), and, further down the page, "Den hellige Ånd":
Den hellige ånd
"Den hellige Ånd" may be a remnant from the time when all nouns were capitalized in Norwegian, more than 100 years ago.
Does this mean that you have seen "Den Hellige ånd"? That is the only combination that seems definitely wrong to me.