Mr Cook died on Saturday at Craigavon Hospital, his family has said. He was in his 70s.
He was a founding member of the Alliance Party in 1970 and was elected to both Belfast City Council and a previous Stormont Assembly during his political career.
Alliance leader Naomi Long paid tribute to his vision, along with those who founded their party.
She hailed his achievements, which included becoming the first non-Unionist to serve as Belfast’s lord mayor when he took up the role in 1978.
“He served the South Belfast electorate in Cromac in Belfast City Council from 1973 to 1985 and then at the ill-fated NI Assembly from 1982-86.
“A committed European, he also stood as an MEP candidate in 1984,” she said.
“His loss in this, our 50th year, is all the more poignant as we never got to celebrate that anniversary together, as a result of Covid-19, to which he ultimately and tragically succumbed.
“My thoughts and those of the Alliance family are with Fionnuala and the wider family circle as they mourn his passing at this sad time.”
Mr Cook was a solicitor and also founded the Northern Ireland Voluntary Trust, now known as the Community Foundation for Northern Ireland.
He is survived by his wife Fionnuala, children Barbary, John, Patrick, Julius and Dominic, granddaughters Romy and Imogen, and sisters Alison and Nora.
