Shirley,
I assume this is your family in 1901 & 1911:
http://www.census.nationalarchives.ie/pages/1901/Antrim/Ballymena_and_Town_and_Urban_District_of_Ballymena/Bridewell_Street/925603/
http://www.census.nationalarchives.ie/pages/1911/Armagh/Portadown_Urban/Park_Road/306356/
Is this the father’s grave in Ballymena cemetery?
IHS Sacred Heart of Jesus have Mercy On the Soul of George Albert Montgomery, Died 24th Nov. 1937, aged 38 years. And of his father George Montgomery, Died 16th April 1938, Aged 69 years.
Probate abstract:
Montgomery George Albert of 37 Princes Street Ballymena county Antrim postman died 24 November 1937 Administration Belfast 14 February to George Montgomery hotel porter. Effects £160.
George junior died intestate and so there won’t be a will but the probate file (held by PRONI) may give you a little information on his next of kin.
The death records for the years 1915 to 1967 are on-line on the GRONI site. But Arthur could still have been alive in 1967. For years after that you need to get someone to go to GRONI or PRONI in Belfast and search them for you. Marriages up to 1942 are on-line on the GRONI site. You could search them to see if Arthur married in Northern Ireland. Post 1942 marriage records are not on-line and also require a trip to GRONI or PRONI.
I looked for a will for Arthur Montgomery up to 1966 but there’s none that fits. However records post 1966 are not on-line, and again require a visit to PRONI.
There aren’t many records to search post 1911, and given that a lot of people moved around for work, it can be difficult tracing them without some clue as to where they had gone.
If you know the family address in 1939 you could make an FOI request to see who was living there in the 1939 Register (which is held by PRONI for Northern Ireland records). But you can only do that if you have an address. If you don’t know where someone was living, you can’t search it by name.