Yes, that could be the problem. A quick way to test it is to pluck the string, then lightly press down on the string just beyond the nut. If that stops the buzz, then you need more break angle to get the string seated down in the nut groove. More wraps on the post, or a string tree.
If pressing the string up there doesn't stop the buzz, then the problem is most likely that the nut slot is worn too low, causing the open string to lightly ring on the 1st fret. The quick fix for a low nut slot is put a drop of superglue and some powder in the slot, then re-file it to the correct height.