Not if the Enterprise were traveling generally either "up"' or "down." The distance to the upper edge of the galactic disc is much less than the distance to the outer most rim.
Except it was referred to as the rim of the galaxy in "By Any Other Name", in reference to events of "Where No Man..." - so it can't have been up or down, as the Star Trek Star Charts
try to fudge.
ROJAN: There is an energy barrier at the rim of your
KIRK: Yes, I know. We've been there.