My mandarin textbook says it means "speak to democracy," but I can't find a definition for this word anywhere on the internet. Even google translate says it is not a word.
It is not one word
讲 (v) speak
民主 (n) democracy
In Cantonese, the meaning for '讲' includes being the shorthand for 讲究 (pay attention to; regard); 讲求 (stress), = 注重; 追求
讲民主 (speak democracy) = regard/ pay attention to democracy
不讲民主 (speak democracy) = disregard/ not pay attention to democracy
讲道理 (speak reason) = reasonable
不讲道理 (not speak reason) = Unreasonable
香港人讲生活，讲享受 = "Hong Kong people pay attention to life and enjoyment"