- That’s Chinese, not Japanese
- Yes, they do just use 碁 (go) to mean the game go. Source: the club I went to in Osaka:
Note they use 囲碁 frequently, such as on the sign on the lamppost, since words with a single mora are short and often have many other meanings as well (in case of go, it also can mean: language, five, a honorific prefix, a prefix meaning mistake and countless pronounciations of kanji in compound words). - If you’re not convinced, it literally states on the intro of the Japanese Wikipedia page that 「 単に碁（ご）とも呼ばれる」(translated: it is also called simply go), there’s this question on stackexchange and finally here you can read it from a native speaker.
But of course I’m just trying to be facetious 
Personally I find the Japanese kanji a lot nicer, 囲 even looks like a go board! And 碁 just means go, whereas 棋 usually is more associated with chess (of which the local variants xiangqi and shogi are more popular in China and Japan respectively, not sure about South Korea though).