Banquo obviously did dream of the Weird Sisters. That is why he says, either to himself or to his son Fleance:
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!
He is afraid to go to sleep again because he feels sure he will have more dreams about the Weird Sisters and what they have promised him. He and Macbeth are both suffering from insomnia because of the Weird Sisters. When he tells Macbeth
I dreamt last night of the three weird sisters:
To you they have show'd some truth
he may be hoping they can commiserate. The three witches have caused both of them serious emotional problems by their promises. But Macbeth has already made concrete plans to fulfill the witches' prophecies as far as they concern him. He doesn't want to discuss them with Banquo for fear he might give himself away. His statement, "I think not of them," is intended to dismiss them as if he thinks their prophecies were nothing but a lot of crazy poppycock and that the fact that they did correctly predict his appointment as Thane of Cawdor was just a coincidence.
But Macbeth has second thoughts. He half-suspects that Banquo might be hinting that he would like to discuss how the two of them might make the prophecies come true. Macbeth could certainly use some help. If he kills Duncan, he still has Malcolm and Donalbain to deal with. Malcolm is Duncan's heir apparent. This may be Macbeth's only opportunity to kill the sons along with their father--but he doesn't know how he can handle three murders all by himself. What he means when he says
Yet, when we can entreat an hour to serve,
We would spend it in some words upon that business,
If you would grant the time
is that he would like to talk to Banquo about the two of them killing Duncan, Malcolm, and Donalbain together. There will never be another opportunity like tonight. The implication seems to be that Macbeth would become king and would then arrange to have Banquo's son his heir apparent. But Banquo doesn't trust Macbeth. If Macbeth is capable of killing Duncan, then he is certainly capable of killing Banquo. And this, in fact, is what Macbeth actually does.
Banquo turns him down. Macbeth has to go it alone. And he botches the job. If he had intended to kill Malcolm and Donalbain, he is thwarted by the voice he imagines crying "Sleep no more!" and then the knocking at the gate which seems to threaten to wake everybody in the castle.
Banquo informs Macbeth:
"I dreamt last night of the three weird sisters:
To you they have show'd some truth."
"I think not of them:"
Macbeth obviously wants to convince Banquo that he does not consider the witches' predictions with any real import and that they were a mere trifle. He does not deem what they have said as worth anything. This is clearly a lie, since he had been enthralled by their prediction and had been most impressed when he became Thane of Cawdor as they had predicted, so much so that he wrote his wife a lengthy letter informing her of their meeting and the accuracy of their prediction.
Macbeth and Lady Macbeth had already plotted Duncan's murder and Macbeth says this to Banquo so that he may not become a prime suspect after Banquo's assassination. If it seems that he does not deem the witches' predictions as anything of value, it removes any possible suspicion that Banquo might have later.
It is ironic, however, that immediately after, Macbeth requests that he and Banquo "spend ... some words upon that business". 'That' seems dismissive of what happened, but Banquo is clearly not convinced, saying that he
"shall keep my bosom franchised and allegiance clear"
He clearly states that he will not, for any reason, betray his king, suggesting that he already suspects that Macbeth is up to something.
Macbeth did not lie to Banquo about the witches predictions in this act, or scene. Banquo tells Macbeth that he has had nightmares about the witches perdictions. Macbeth responds that they haven't bothered him, but they could talk more later. Macbeth in truth had been bothered about the witches. You can find more to this in the link below.
As soon as the witches give their predictions about Macbeth being king, Macbeth begins to think of little else. However, in Act 2, Scene1, he encounters Banquo. Banquo mentions the witches but Macbeth lies and says, "I think not of them" (Act 1, Scene 2, Line 21). In truth, he has been thinking all about their prophecy and has even been discussing it with Lady Macbeth. The reason he lies is to throw Banquo off and hope that Banquo will be suspect Macbeth if Duncan dies. Macbeth even goes so far as to suggest that he and Banquo could work together. Banquo replies that is fine as long it doesn't tarnish his honor. So Macbeth quickly wishes him a good night.