You haven’t mentioned your partner’s father’s name which makes searching for his death tricky. However deaths in Northern Ireland in 1964 are on-line (they go up to 1970) and you can look it up on the GRONI site using the “search registrations” option:
https://geni.nidirect.gov.uk
You will need to open an account and buy some credits. It costs £2.50 (sterling) to a view a certificate.
I assume that Florence McCormick married Mr Loughran around 1955 in Northern Ireland. If so, that marriage certificate isn’t available on-line. It is open to the public just not on-line. You need to contact GRONI and get them to search for it for you. The marriage certificate should give you both their fathers’ names and occupations and from that you can search back to their births, and sideways to their siblings births. Births within the past 100 years are not on-line and again you would need to ask GRONI to search for them. (Or you can make an appointment with GRONI in Belfast and do it yourself, or get a researcher to do it for you).
Contact details for ringing or e-mailing GRONI:
https://www.nidirect.gov.uk/contacts/contacts-az/general-register-office-northern-ireland
Phone: 0300 200 7890
With the fathers names and occupation you may also be able to search for a will on the PRONI wills site. I did search for Loughran wills in the Dungannon area in 1964 but the only one was a farmer married to Winifred, so presumably not your family:
Loughran, Patrick of Camaghy House Cappagh Dungannon county Tyrone farmer died 20 June 1964 at South Tyrone Hospital Dungannon Administration Londonderry 18 May to Winifred Loughran the widow. Effects £2117.
I didn’t see a Florence Loughran death up to 1970 so presumably she died after that. But again you can ask GRONI to search for it (there may be search fees). With a date and place of death you could then get a researcher to look for a will in PRONI. (Probate abstracts after 1966 are not on-line). Florence’s death certificate may also give you a lead to any next of kin.
You probably need to engage a researcher in Belfast to do some of this for you.