Yes. Cicero did mention his grandson in very few letters that we know. But we do not know very much about this child, except that it was born away from its father Dolabella, who by then went by the name of Lentulus, after an adoption of his into that plebeian family, to get into the political office of tribunate. A short bout of letters and no further mention of the child led to the conclusion that it also died very soon after birth, shortly after its mother, early in 45.
Tullia’s child was also a son and was born safely, some time in January. Asconius (5C) tells us the birth took place at Dolabella’s house, but he does not mention the divorce, so he may be making an assumption. It is more likely that the baby was born at Cicero’s house. […]
The baby was called after Dolabella and presumably recognised, by letter, as Dolabella’s younger son. Tullia, probably some weeks later, was taken to the villa at Tusculum to convalesce. She died there in February, no doubt of complications. We do not know where she was cremated (perhaps in Rome) or where the ashes were deposited. She was perhaps thirty-two. […]
There are no letters to Atticus in December, January (both Cicero and Atticus were in Rome) and February. The correspondence resumes on 7 March, when Cicero had hidden himself away at Astura. So we have no details on Tullia’s illness and death. Cicero grieved long and sincerely. […]
Tullia’s baby was known as Lentulus (his father’s new name). Cicero wrote to Atticus:
‘I should like you some time, when it is convenient to you, to visit baby Lentulus and choose slaves, as you think fit, from the staff and assign them to him’.
It sounds as if the baby was at Cicero’s villa at Tusculum (rather than in his house in Rome, where it would have been easy for Atticus to drop in), for it appears the baby is in the same place as the household slaves, who seem to be Cicero’s own, perhaps including those who had loved and served Tullia. A wet-nurse (unless Tullia suckled him herself as long as she could) and other staff must have been found for him at birth.
These are extra attendants of some sort, perhaps a doctor or a male paedagogus, for instance, or child companions, and possibly they are to be transferred to the baby’s ownership. In any case, we see Cicero taking a close interest in the baby’s welfare. There is a follow-up letter thanking Atticus for visiting him and asking him to assign whichever servants and as many of them as he decides. Unhappily, this is the last mention of the child, who probably died in infancy.
— Susan Treggiari: "Terentia, Tullia and Publilia. The Women of Cicero’s Family", Women of the Ancient World, Routledge: London, New York, 2007, p139.
The letters mentioned are classified as follows (formatted as quote, but shortened from the endnotes):
A 12.28/267.3, Astura, 24 Mar. 45.
The child would eventually have been sent to his father’s house to be raised. Cf. A 12.33/269.2, Astura, 26 Mar. 45, for the servants surrounding Attica.
12.30/270.2, Astura, 27 March 45. If they were made part of the baby’s ‘property’ (peculium, the possessions a son-in-power had for his own use), they would in law belong to Dolabella.
A 12.30/270.1, Astura, 27 Mar. 45.