A few corrections are in order:
Yes, the statement is true, but the answer above has it a bit backwards (probably just a proofreading error): Macbeth orders the murder of both Banquo and his young son, Fleance. The murderers are able to complete only half of their job: they succeed in killing Banquo, but, in the dark of night, Fleance escapes.
This all takes place just before (or even during) a formal dinner that King Macbeth is throwing for himself in his castle.
Furthermore, Macbeth does not think that he will ever have any children, so the attempted murder of Fleance is done out of jealousy of what had been promised to Banquo by the witches.
This statement is true. Macbeth does have Fleance killed (and tries to have Banquo killed) for this reason.
In Act I, Scene 3 of this play, Banquo and Macbeth meet three witches. These three witches foretell that Macbeth will be the Thane of Cawdor and the King of Scotland. They say that Banquo will not be a king but will be the father of kings.
Later on in the play, Macbeth has become king by killing King Duncan. He now decides that he should kill Banquo and Fleance so that his own sons will be kings after him.
The statement is very true. The new king Macbeth is supposed to order his men to kill Banquo along with Fleance since he was prophesied by the "weird sisters" that Banquo's progeny would surely succeed to the throne.
*** And, this is not mentioned anywhere in the play that Macbeth would never have any child.
Yes, Macbeth is jealous indeed. But it is not only his jealousy which drives him to give order to kill Fleance. This is because of his inert feeling of insecurity which is the main reason. He is a highly ambitious person, and can easily exterminate those who would be like a threat to his holding on the royal power. This is why he orders to kill the son of Banquo, Fleance.
True. Fleance is Banqou's son, so Macbeth feels murdering him also will help inhibit the prophecy. Which is why Macbeth speaks of the faliure to kill him as not a current problem, but it will eventually be.