Thanks again, Elwyn.
My grandfather wrote the name with an apostrophe. My father wrote the "c", but raised above the line. But I was surprised that irishgenealogy.ie didn't give results for the variants when I searched McDowell; someone else alerted me to the death certificate for John McDowell of High Street Antrim.
I think I have "found" the graveyard (so far as that makes sense from several thousand miles away) since my previous post. In a different context I came across a reference to the Seneschal of the Manor of Moylinny, and I thought that looked plausibly like a different version of the Irish name. It turns out that Moylinny is a townland in Antrim town; surely likely the same as Moylena where the beetling mill was. And a search for a Quaker graveyard in Moylinny turned up a result on genuki.org.uk:
https://www.genuki.org.uk/big/irl/ANT/Antrim/QuakersGraveyardCemetery
It shows the site S of Belfast Road, close to Nursery Park.
I found a couple of photos (from 2007) on flickr.com:
https://www.flickr.com/photos/21067190@N00/775794563/in/photostream/
One shows a handsome stone gateway with an iron gate, through which some gravestones are visible. The other is taken from close to the gate, showing gravestones and an area enclosed with low iron fencing.
I also found a document (from 2000) about the heritage of Antrim town, in which the gateway and walling of a Quaker graveyard at Belfast Road, Antrim appear on a list of listed buildings:
https://www.nienvironmentlink.org/cmsfiles/Antrim-Heritage-Audit-v2.pdf
From what you say, the gravestones visible in the pictures are likely to be of non-Quakers. But I don't know that John was a Quaker; the only evidence of that, so far, is the death notice in the Belfast Newsletter. I've contacted the cemeteries department of the government of Antrim and Newtownabbey asking if they maintain that graveyard, and suggesting that it might be a good idea to have some organization like Find a Grave record the gravestones. And I will contact the Provincial HQ in Lisburn as you suggest.
John
P.S. I copied and pasted the web addresses, expecting them to look like links, and they don't. I hope they work for you. (I'm assuming that as a resident in the area unfamiliar with this graveyard you will be interested in it independently of helping me.)