According to the page on liberties the word qi is the chinese word to describe liberties, they are the breath of the piece, it’s life-force. It’s mentioned that this qi is the same qi that is referred to in Feng Shui. I would conclude from that statement that it is also the same qi in Tai Ch’i - as Ch’i is another way to spell it.
So what is the relation to the Qi in Wei Qi? With my limited understanding of Chinese it appears that the symbols are totally different, yet I find it strange that the idea of qi (life force) is so closely related to the game and shares a similarity with the title of the game by coincidence.
Although 棋 and 气 are both romanized as qi, their tones are different: 棋 is second tone (qí) while 气 is fourth tone (qì). And once the tone is changed, we are most likely talking about different characters, with most likely different meanings.
I'm not sure how many Chinese will associate between the two. To me, I draw no relation at all from the language perspective. It just happens that 气 is a key concept in 围棋.
unkx80 is dead on correct. In Chinese, the pronunciation and the tone together is what gives you meaning; either without the other makes no sense. The fact that 气 (life force) and 棋 (chess) are both romanized 'qi' is like a false cognate.
As a side note, qi in tai ch'i (ch'i is the old spelling for both qi and ji) is 极(極), which has yet another different meaning.
Thanks to you both, that is very interesting!