No, a game is only over when both sides concede, and players concede when they recognize that any move they could make either does not change the score or would cost them a point (playing in territory that is already yours effectively reduces your territory as only empty territories are counted as points (in standard japanese rules)).
If one player would never concede you would keep playing until the board is full except for the unfillable eyes of the player that wins. Obviously that would take a long time and many negative moves of the losing player.
So you stop playing when both players are confident the game is over and agree on who has won.
That problem of "negative moves" is solved by the Chinese rules. A point is scored under them for each surrounded intersection plus each stone on the board. So you can defend any supposed weakness without losing points (you still lose sente and points if you needlessly defend when the score is not settled yet). You can even reduce your territories to two eyes each if you feel like.
By the way, the Chinese rules let solve neatly all those cases handled by special rules in the Japanese rules. The bent four in the corner is the most notable one. Playing it out with the Chinese rules is a neat explanation why that corner is defined to be dead: the surrounding player defends any weakness without losing points and starts the ko. The other player has no ko threats and dies. Those defensive moves lose points under the Japanese rules so they have to make a special rule for that shape and many others.
The only problem with the Chinese rules is that scoring takes longer and completely destroy the shape of the game: you fill in the territory of one player, take out the other's stones and count by grouping the stones in convenient shapes. Furthermore if you want to count during play you must remember how many stones have been captured because prisoners are returned to their bowl and are not stored in plain view (the score penalty is paid by not having those stoned on the board). Japanese rules are a shortcut that makes scoring easy but the tradeoff is the dictionary of special cases at the end of the game.
"Conceding" means passing. If you don't pass, you have to play. If you play when the game is effectively over, you either play in your opponent's territory and get captured, or your own, which reduces your score. If you play in your own territory enough then you can actually end up losing your eyes, and then be captured.
I guess the remark comes from the "perhaps infinite" proof game. I don't know why they put "perhaps infinite" there as most Go rulesets have rules against repeating board positions.
A mathematically established lower bound on the longest possible go game is 10^10^48 [1]. While this isn't infinite, it's certainly large enough to be considered infinite in practice.
For what may or may not be an example, Graham's number is astonishingly huge, but math world (http://mathworld.wolfram.com/GrahamsNumber.html) claims the answer to problem it is an upper bound for may even be larger than 11:
"Graham and Rothschild (1971) also provided a lower limit by showing that N must be at least 6. More recently, Exoo (2003) has shown that N* must be at least 11 and provides experimental evidence suggesting that it is actually even larger."*
In this case, I think it is safe to claim that the answer is at least 361!/8, though (but that may already include many truly silly games with suicidal moves in the opening or games that continue way past the time experienced players think they are over)
A computer would only win by playing out pre-proved positions (joseki) against an amateur who has studied them positions without understanding them.
As for infinite games, they don't really happen much; in rulesets that make it possible for them to happen, the game is usually called "no result", and this has happened only very few times in hundreds of thousands of recorded games. Modern rulesets have "patched out" this by implementing superko or similar rules: playing a stone that would put the board in a state it was in previously (positions of the stones and which player's turn it is) is an illegal move.
For each of the (several) traditional rulesets there are positions that the ruleset can't deal with. So in order to allow a mathematical or computer analysis the rules often need to be tweaked.
If true, does it mean a computer (or an Human) could win by just not conceding?