Is it true that deceased children do not intercede for their mother unless an 'aqiqah is sacrificed on their behalf?

What is mentioned in the question is correct. It has been narrated from Imam Ahmad ibn Hanbal that the meaning of "every child is a pledge for his aqiqah (definition)" is that if a child dies and no aqiqah is offered for him, he will not intercede for his parents, or that the aqiqah is binding upon him, like a pledge that cannot be released except by its fulfillment. The aqiqah is a confirmed Sunnah (definition) according to the majority of scholars, and obligatory according to the Zahiris and in one narration from Ahmad. It is recommended for the father to offer aqiqah for his children to gain reward and to avoid disagreement.