I agree. It should be 'Quite honestly'. This is a common expression used before giving an opinion on something, especially if it's a strong opinion. 'What did you think of xx?' 'Quite honestly, I thought it was a complete disaster.'

I think you might have misunderstood that though quite honestly does exist and means very honestly

quite adverb 1. to the utmost or most absolute extent or degree; absolutely; completely. "it's quite out of the question" synonyms: completely, entirely, totally, wholly, absolutely, utterly, thoroughly, altogether very; really "“You've no intention of coming back?” “I'm quite sorry, but no, I have not.”" 2. to a certain or fairly significant extent or degree; fairly. "it's quite warm outside" synonyms: fairly, rather, somewhat, slightly, relatively, comparatively, moderately, reasonably

I don't think there is such a thing. I quite honestly think that.... It is not "quiet" it means silence but quite has a different meaning in the sentence you asked. so quiet and quite are different words.