Not all intelligences would need to be biological, either. The point is raised above that computers may well become intelligent (depending on what your feelings on hard AI are, you either do or do not believe this) sometime soon, so that would be an example of non-biological intelligence. But that's a little bit of a tricky case, as I noted, because intelligent computers on Earth would require oxygen-breathing designers, unless there is an eensy-weensy sulfur-reducing bacteria civilization that we're overlooking that's created true AI.
However, this doesn't rule out (depending on the hardness of your sci-fi setting) the existence of organized forms of matter that are not organic (by our standards) becoming intelligent. A good example of this are the Qax from Stephen Baxter's Xeelee cycle--they're definitely intelligent, but because they're essentially a kind of standing waveform in chaotic fluid systems, they're sure not biological. And they do not need oxygen to be intelligent, just a system where their thoughts can be encoded. (There's also the photino birds and a whole bunch of other non-biological "organisms" in the Xeelee cycle, so I'd recommend that as reading for people interested in deep-space "life".)
So while oxygen might be necessary for complex, intelligent biological life--and we don't know this, having not checked for eensy-weensy civilizations involving sulfur-reducing bacteria, or theoretical interstellar civilizations involving sulfur-reducing bug-eyed monsters--it's not necessary for intelligence per se.