Edge the area, pull out the grass that is growing in the hedges and remove the sod from the area within a foot of the bushes. Then put down several layers of newspaper over the grass within the edged in area and top off with mulch.

No more grass unless you have some mutant variety.

Round up.....to the "handshake of death"...put on a plastic glove and put a cheap cotton glove over the plastic one....wet the glove with round up and "shake hands" with bits of the bad grass.
This will allow you to be careful with your other plants and still kill the grass.
Linda C

It's safer -- for you -- to use a paint brush rather than the sort of gloves suggested.

If the hedge isn't very foliage-dense then enough light will get through to let the grass grow through. Can't be helped with a deciduous hedge, I know.

In addition to the paper or cardboard exclusion try feeding the hedge with a good layer of mulch. That boosts the foliage output and also changes the quality of the soil.

I know when I've done this a couple of things happen for me: the soil becomes easier to work and I can pull out the offending weeds; and they often don't grow back because the soil is too good for them.

If you have quantities of lawn clippings and you've cut before seed heads form - put the clippings under the hedge leaving about a couple of inches clear around the trunks so they don't go mouldy. No more than three inches thick otherwise it goes slimy and gets a waterproof crust on top which sheds the rain.

If you do this in autumn when you could have dry leaves mixed with the clippings it will make a useful weed suppressor for over the winter. Summer grass clipping mulch does need stirring up to break that surface crust and let the air/rain through.