He was born during the lifetime of the Messenger of Allah ﷺ, and the Prophet performed tahnik for him (1).
He is the very child whom Umm Sulaym conceived on the night her previous son died. She kept the death from Abu Talhah until he had finished his supper; then she beautified herself for him until he came to her, and she conceived this boy.
The next morning Abu Talhah went to the Messenger of Allah ﷺ, and he said to him, “Did you spend the night together as newlyweds? May Allah bless the night you shared (2).”
It is also said that the deceased child was Abū ʿUmayr, the little boy who kept the pet nughayr (3).
ʿAbd Allah grew up, pursued knowledge, and was blessed with ten sons who memorized the Qur’an; most of them also transmitted learning. Among them were Ishāq ibn ʿAbd Allah ibn Abī Talḥah—the teacher of Mālik—and ʿAbd Allah ibn ʿAbd Allah.
Those who narrated from him include these two sons, Abū Tuʿālah, Sulaymān the freedman of al-Ḥasan ibn ʿAlī, and others.
He himself transmitted only a small number of hadiths, all of them on the authority of his father and of his half-brother Anas ibn Mālik.
(1) Reported by al-Bukhari 9/508 and Muslim 2144: Anas said, “I took Abdullah ibn Abi Talhah al-Ansari to the Messenger of Allah ﷺ when the child was born. The Prophet was wrapped in a cloak and was rubbing tar on one of his camels. He asked, ‘Do you have any dates?’ I answered, ‘Yes.’ I handed him some dates; he put them in his mouth, chewed them, then opened the baby’s mouth and placed the chewed dates in it. The infant began to smack his lips, so the Messenger of Allah said, ‘The Ansar love dates.’ He then named him Abdullah.”
(2) For the full version, see al-Bukhari 3/135, 137 (Book of Funerals, chapter: “One who does not display excessive grief at a calamity”) and 9/508 (Book of Aqiqah); also Muslim 2144 §23 (Book of Etiquette, chapter: Recommendation of tahnik—chewing a date and rubbing it on the newborn’s palate).
(3) Al-nughayr is the diminutive of nughr, a small bird resembling a sparrow with a red beak (pl. nughran). Anas ibn Malik related: “The Messenger of Allah ﷺ had the finest character. I had a younger brother called Abu Umair—I think he had already been weaned—and whenever the Prophet came he would say, ‘Abu Umair, what happened to the nughayr?’—referring to a little bird the child used to play with.” Reported by al-Bukhari 10/481 and Muslim 2150.