"Don't talk to him about God; talk to God about him."
— My Christian mother about my atheist father
In anything with love for others, whether romantic, family, or friends, we often want to tell others about our positive hopes and feelings, but love knows when to shut up.
Just respect your coworker. Don't even declare your choice to shut up about God. If you want to pray for anyone to have any kind of happier life, by all means do it!
But, that's the thing about what prayer claims to be...
Supposedly, prayer makes a difference. If we have to tell others we pray for them for prayer to work, then prayer doesn't really work. If you believe prayer makes a difference, then let prayer work how it claims to: between you and God. If it really works, you won't need to tell people what you pray for because it will happen without saying so.
How can I declare my effort to help with prayer, without offending him?
Just don't declare it to him. That is badgering. As Edith Bunker said to Archie, "[If it's God's business,] then you let God handle it."
Just a few days ago, my mother and I were talking about how we can't badger people into changing. We recalled and shared stories from dad, how he taught us as Christians while he was still an adamant atheist. We have lots of friends and family, Christian and non-Christian, who have money troubles and religious questions of all sorts. We only want them to be more responsible with money and be nicer to others as a reflection of them finding peace in their hearts. Mom and I have both learned, as we just discussed: Pray silently, love others, respect them, and be glad for any small way anyone finds more happiness.
I am a Christian, my father was atheist/agnostic (he himself said he changed through his life, I understand the difference). 8 years before he died, he had his own religious experience all by himself and became a "Christian" in his own way, on his own terms, not the result of us proselytizing and pestering.
Not that this is a goal, but we must love people as we let everyone find and choose their own way.
Currently, I am in Asia with family friends, a bordering-depression/suicidal teenager who became Christian on their own and approached me for Christian teaching (bec. I was a Bible major in college), who has a nihilist father (sometimes called atheist or agnostic by his family). The mom and dad love their teenager, but all these worldviews make things a little complex...
In talking with the teen regularly, I address three ideas from Christian-Bible tradition. (skip if boring) 1. "God loves us, thus we have value", 2. "Love others as yourself, then you will be happier", 3. "Get deep satisfaction by pressing-through-hardship, not despite it". Those help the teenager be happier and more stable; both the mother and father thank me at times for helping with the improvement in their teenager (everyone is helping everyone, it's complex). But, the father, my friend, is concerned that my conversations are too Bible-heavy. I myself strongly believe in arguing for both sides of a debate, like "nihilism-to-atheism v Christian-theism" because a belief can only help us if we really understand all sides of it—and I want to help everyone for real, not just make minions who parrot what I say while still depressed inside. But, if I try to explain that to the father, my friend, he doesn't want to listen and ends that specific conversation. I only guess it is because Christians have a well-earned reputation of "trying to sell Amway" so to speak. But, I'm not; and I don't "go to church", so I am not trying to "rack up convert points". I just want to help a family friend with what a. I am trained in from college and b. has been working in this situation. But, I can't say that to the father, my friend, without him feeling pushed.
...That's our friendship conflict.
For me with that father, I don't talk about any touchy topic unless asked. Even with the teen about Christian teaching, I never say, "This belief is true," but, "This is the Bible's opinion/teaching; you don't need to agree." (I say that almost every time.) I am very big on not brainwashing people about religion. Even if I believe something is true, I do not decide that for anyone else. I just represent myself when asked and BE a Christian in my work and social conduct. Truth is truth, but no one knows truth fully. And, the important thing about truth is that we each recognize truth on our own without being pushed, otherwise it's pointless.
Sometimes, the father, my friend, gets a little offended and upset, misunderstanding what I say...
I just leave it be and keep respecting him.
We remain friends and only become better friends because I don't try to "fix" him. I want to let him know the truth about my actual opinion about a few things, but that would be pushy. Respecting his desire to not talk about religion is DOING the "Christian" thing (for my part) and not just badgering about it.