Ashers bakery, which is owned and run by the McArthur family, has been taken to court by the Equality Commission after it rejected the order of a gay rights activist on religious grounds.
The bakery said it could not make the cake, featuring Sesame Street starts Bert and Ernie, because it believed it to be unbiblical.
The Christian Institute has backed the bakery and is paying its defence costs.
Ashers manager Daniel McArthur, 25, said the court process had been "difficult and exhausting".
He added: "Our faith is very important to us; it determines how we live, how we bring up our children, how we run our business, how we meet and how we engage with other people in society, so yes we can't leave it out whenever we go to work in the morning.
"It's been a difficult and exhausting time for us as a family but God has been faithful to us. And He has given us the strength to deal with this, and we know and trust in Him that going forward He will continue to give us His strength."
He added that his wife Amy had been at this side throughout the whole court case.
"We depend fully on God, and God gives us our husbands and our wives in these difficult times to help us through them.
"There's been many parts of the Bible which I've read which have been of great encouragement, but particularly Psalm 22.
"When I'm reading it I'm thinking about what Jesus has been through, and what He's been through for me and for our family, and I know that if He's been through that, God in turn will give us the strength to deal with the trials which we have to go through.
"We do want to say thank you to the thousands of people who have supported and prayed for us, we're very grateful and certainly your prayers would be much appreciated as we move forward.
"We also would like to thank The Christian Institute for their help during a difficult and exhausting time for us and our family, and we're thankful to God as well who has been faithful to us through it, and we know that He'll continue to help us bear this burden as we go forward in the future."
Christian Institute Deputy Director Simon Calvert said: "The McArthurs, like countless other Christian business owners across the country, simply want to live and work in accordance with their Christian beliefs – beliefs which have long had a positive influence on Western culture and history."
Gareth Lee, the LGBT activist who ordered the cake, has not spoken publicly about the case.
But in court he told the judge Ashers' decision "made me feel I'm not worthy, a lesser person and to me that was wrong".
In a statement the Equality Commission said: "The Commission's policy is not to discuss the details of any case we are supporting while the matter is under consideration by the court."
At the time of the incident the public organisation said: "The Equality Commission has an important role in ensuring effective application of Northern Ireland's equality laws and supports cases so that people are aware of, and can avail of, the protection these laws afford against all forms of unlawful discrimination.
"This case raises issues of public importance regarding the extent to which suppliers of goods and services can refuse service on grounds of sexual orientation, religious belief and political opinion."
