In a short story I'm writing, I have a communication system between a group of characters which, in essence, allows telepathic conversation. So I want to present actual dialogue using this system, only it's non-verbal. What's the accepted way to style this with regards to quotation marks?
The same way you would render dialogue over a telephone, or radio, or signing, or morse code. The medium is unimportant. You treat it as dialogue, period. So you use either quoted literal text, bock quotes without quotation marks, or unquoted indirect text (i.e. you are conveying the sense of what was said rather than the literal phrasing). Examples: Note that the block quote is usually reserved for longer passages. Read He said, she said - Mechanics of Dialogue