So make the ice breaker something about praising and respecting your chosen God.
If they're a dark, evilly thing - do it over a slain foe. Or bring them something suitably evil for their altar.
If they're a lightwalker like Corrlaan, consider humble prostration as a casual act just to garner some attention.
If it's a neutral like Rayihn, go for some vanity-feeding praise about her rumoured beauty and benevolence. And so on.
Once the imm has their eye on you using a basic approach technique, they'll know to watch for good religion-style behaviour. Then they'll start rewarding it. And the next thing you know, it's tea and crumpets with Zulgh. Just use an ice breaker, or a simple note stating you 'wish to learn more' or 'i am devoted to your path'. Then take things naturally from there.
Yhorian