Was able to gain desired output.
Work around:
1.) Create two grids
(see the numeric boxes on the snip above)
2.) Set formatting for grid.
Labels 1 & 3 are visible for the X-Axis Grid.
Labels 2 & 4 are visible for the Y-Axis Grid.
(Make sure to turn off the edges that would cause overlap by unchecking the visible box)
Still cannot get the grid to be visible for the full extent of the quarter section grid, but this work for using the map series option in layout view. So this does not fully compensate as work around solution, based on my experience thus far.