It's cock. Took a course with Andrew Appel; the heroic man did not flinch once when he said it over and over again during the semester. My favorite line of the course: "You're going to do your homework in COCK."

Was the naming intentional? Did they know the common English meaning of it?

They give an explanation here.

https://coq.inria.fr/faq?q=node/16&som=2#htoc4

Reasonable people might disagree, but I think that what they say there doesn't answer the question at all. Cock means rooster in English too, after all, but somehow the name doesn't get used.

It's actually pronounced more like "cock". Coq is the French word for rooster.

In a french accent it's more like "Cuck", that's how I say it in my head. Luckily, I don't have to say it out loud.

I believe it's /ɔ/ in French but usually /ɑ/ in English.

Of course, cock is an English word for rooster.

Does coq have the same connotation in French, though?

Not at all, but I was told that the pun was totally intentional [citation needed].

Is the filename of "coqasm.pdf" also trying to get at something?

Hrm.... Really makes me uncomfortable to say "I love Coq" conversationally.