Depending on the context, the three phrases could be used in similar situations, but they mean slightly different things. The phrase 'no reason' is actually different from 'a random reason'. In the second phrase there might be a reason, but it doesn't have anything to do with the expected reason, whatever that might be. So those two could mean different things.
The phrase 'at random' has a sense of describing the timing of the speech more than the reason for the speech. For instance if the speaker talked about his dog at a time when it was not expected, or in between other unrelated topics. His behavior might be similar to what's being described by the other phrases, but the narrator is focusing on a different aspect of the situation - the timing rather than the reason.
Some other similar phrases are
He started to talk about his dog out of the blue.
He started to talk about his dog suddenly.
He started to talk about his dog with no notice.
He started to talk about his dog spontaneously.
- 'no reason' means the speaker starts talking without any purpose
- 'a random reason' means that the speaker starts talking for a reason that doesn't follow logically from the rest of the situation
- 'at random' means the speaker starts talking at unusual or unexpected times