I guess the Polish sentence isn't clear about whether it's the closest June, or June in general. But June in general seems more probable. To make it absolutely clear, you could say "W czerwcu nie pijam kawy", using the habitual verb "pijać". The habitual verbs are outside the scope of the course, because generally, the normal verb is enough, especially for a non-native. There are not many of them in common use anyway.