I'd say you have a start. The trick is done before the conversation ever starts, at least for one of the characters.
I'll presume "strong feelings" is a euphemism for love or friendship. Also I am not sure if the person granting the favor is male or female. So let's call them Joe and Charlie (a 50/50 M/F name), Joe needs a favor, Charlie can grant it (maybe).
Generically speaking, you need to build the relationship ahead of time, for at least one character and if possible both of them. Obviously Joe has heard of Charlie, so you can start with him: He hasn't just heard of Charlie, he has heard good things, admires Charlie, likes what Charlie has done and thinks Charlie is a good person. Joe will approach Charlie with high expectations and low suspicion.
Joe doesn't have to be loved at first sight by Charlie, in fact it is more plausible if Charlie is reluctant to do the favor. So Charlie asks for details, and about Joe's motivation, why Joe is asking instead of his friend, Charlie wants to know why they would do any favor. Joe is already primed to think Charlie is a good person, he tells the truth and doesn't hold back.
It is this openness and vulnerability of Joe, along with the reasons for the favor, that resonates with Charlie; making her/him sympathetic. Charlie tells her/his own story, and this resonates with Joe. Even if the favor is not done (or it is not possible to do it), it is the exchange of personal details, and perhaps in Joe's reasons for asking the favor elements of heroism and altruism, and Charlie's sincerity and sympathy, that captures each and makes them feel for each other.
If there is physical attraction as well, all the better. But even if your goal is just friendship, positive strong feelings should make Charlie want to help Joe and hear more about him, and Joe should feel like his beliefs about Charlie were confirmed and better than he thought, she/he is a good person he wants to learn more about.
Writing Advice: This will probably be a long conversation; don't make it two talking heads for a chapter. I suggest you invent a setting for this to take place while they are moving and other things are happening, walking on the wharf or beach, walking downtown, playing golf. Even with one of them cooking. You want to avoid a "white room" effect, during long conversations we want to keep our characters grounded in a setting that can keep the reader's sense imagination engaged, things to see, hear, smell and sense (like temperature, humidity, breeze, etc).