There are several things in Shakespeare's Macbeth that make me think that Banquo is suspicious.
We have some foreshadowing when Ross (in Act One, scene one) brings Macbeth his new title, the Thane of Cawdor, which the witches predicted. At this point, Banquo says (in an Aside):
What, can the devil speak true? (107)
Soon after, Banquo notices that Macbeth is preoccupied after this news:
Look how our partner's rapt. (142)
And then as they leave, Macbeth asks if he can speak to Banquo later about what has happened. It would not take much for Banquo to string the significance of these events together.
This, then, sets the stage for Banquo's concerns—remember also that he is the only person besides Macbeth to hear the witches' prophecies for his friend. (Ultimately we know that Banquo becomes a "loose end.")
In Act Two, scene one, Banquo has misgivings, though at the time he seems not to know where they are coming from; Banquo is very tired, but he is afraid to sleep and asks God to take away the evil thoughts that are causing him to be so anxious—which he fears will set in fully when he sleeps:
A heavy summons lies like lead upon me,
And yet I would not sleep. Merciful powers,
Restrain in me the cursed thoughts that nature
Gives way to in repose. (6-9)
Macbeth tips his hand a little when he tells Banquo that to support him when the time is right will work in Banquo's favor. Banquo agrees, but only if he does not compromise his sense of honor or integrity in doing so. Banquo makes his position clear, though he probably does not yet know why Macbeth would ask such a question—but Macbeth knows exactly what to expect of Banquo.
By the start of Act Three, scene one, Duncan is murdered and Banquo is beginning to go over in his mind their meeting with the witches and what they told Macbeth. As Malcolm and Donalbain fled upon their father's murder, fearing for their own lives, the throne comes to Macbeth more easily than he had planned, for Malcolm had been named Duncan's heir (as the Prince of Cumberland) and Macbeth thought he would have to dispose of Malcolm as well. Banquo notes that Macbeth has it all now, and voices (secretly to himself) his belief of what has happened:
Thou hast it now, King, Cawdor, Glamis, all.../ and I fear
/ Thou play'dst most foully for't. (1-3)
It is as Banquo rides off that Macbeth meets with the murderers he has convinced are suffering because of Banquo's actions (a lie), and he arranges for Banquo and Fleance's murders. It is not just that Banquo knows of the predictions, but the witches said he would father a line of kings, and Macbeth cannot bear the idea that he has lost his immortal soul, not for his own son (he has none), but for Banquo's descendants. And so Banquo is murdered, Fleance escapes (so the prophecy will come true...he will be the ancestor of James I, for whom Shakespeare wrote this play), and Macbeth descends further into a madness that will soon destroy him completely.