It's pronounced closer to "coke".
This is certainly the only pronounciation in the French formal methods community, where Coq comes from. It is also the standard pronounciation in the international formal methods community. I have never heard anyone pronounce it like "coke".
See for example Adam Chlipala at about 5:40 here: https://www.youtube.com/watch?v=4DYVJdHMV5k Benjamin Pierce has a slightly longer o here, but it doesn't sound to me like it's deliberately pronounced differently from "cock": https://www.youtube.com/watch?v=KKrD4JcfW90 And here is Andrej Bauer: https://www.youtube.com/watch?v=COe0VTNF2EA
I know from rare comments like yours that some prudish Americans pronounce it "coke", some deliberately, others apparently because they have been fooled by the first group. But this seems to be a very small minority.
Yes, but...
> which has a short o like the English "cock"
No, it doesn't. The sound it has isn't a common English vowel sound at all; it is closer to the vowel sounds in “cuck” and “cook” than the one in “cock”, but not exactly either of those, either. In fact of the four (those two, plus coke and cock), the “o” in “cock” is the one most Americans would find it most distinct from, I would think.
> https://forvo.com/word/coq/#fr
Yeah, none of those pronounce it with the vowel sound in English “cock”.
> I know from rare comments like yours that some prudish Americans pronounce it "coke",
It's not prudish, its a common way Americans that haven't mastered French pronunciation approximate the unaccented French “o” irrespective of context.
I was mainly talking about the vowel length. However:
> the “o” in “cock” is the one most Americans would find it most distinct from, I would think.
Really? More distinct than from the "oʊ" sound in American "coke"? (IPA from https://en.wiktionary.org/wiki/coke)
It's clearly because I don't pronounce them well, but it's certainly how I do it.
I'm pronouncing it like "coque" from now on.