Lennox's suspicions are not exactly overtly discussed. But it is clear that he is suspicious of Macbeth in his discussions with a Lord in Act 3, Scene 6.
Recall that Banquo is killed after Duncan is killed. Recall that Macbeth claims to have killed Duncan's killers. Therefore, if Macbeth had been telling the truth, Banquo would have been killed by someone else. This leaves one of two possibilities: 1) Duncan's guards killed Duncan and then some other murderer killed Banquo or, 2) Someone else is responsible for killing Duncan, his guards, and Banquo.
Lennox is also present at the feast (Act 4, Scene 3) when Macbeth hallucinates, seeing Banquo's ghost. At one point, Macbeth tells the ghost, "Thou canst not say I did it. Never shake thy gory locks at me." (3.4.49-50). Lady Macbeth tries to convince Lennox, Ross and other attendants that Macbeth is not feeling well. But such a statement (to a hallucination, no less) would certainly confirm Lennox's already growing suspicions.