Information could be encoded in molecule oscillations and therefore 2 rocks of different temperatures would communicate it when touching (heat transfer would occur). Can this be considered processing?
It's doubtful you'll be able to make such a scheme work: Information processing tends to involve storage and retrieval of data. That's not really possible in the thermodynamic regime. You'd have to do all your processing before the information gets lost to thermalization.
Image recognition needs no storage or retrieval of data: it's a single pass thru a series of matrix multiplications. Yet, image recognition is the very definition of data processing.
How do you propose to perform calculations without data storage (think registers)?
Heat is entropy, but entropy is basically a measure of disorder/randomness. So when two substances transfer heat, they necessarily are not communicating information, except perhaps that there exists a temperature differential.