Short answer
Yes. Children of Muslims who die before puberty are in Jannah by the agreement of the scholars, and they can be a cause of their parents entering it. For children of non-Muslims scholars differ, and many leading scholars hold they are in Jannah too.
Muslim children who die young are in Jannah. On the children of non-Muslims, Imam Abu Hanifa is reported to have withheld judgment; many later scholars, al-Nawawi among them, called the view that they are in Jannah the sound one.
The four schools agree on this.
Explanation
The scholars of all four schools agree that a Muslim child who dies before puberty is in Jannah. When the Prophet's ﷺ infant son Ibrahim died, he said that Ibrahim has a wet nurse in Paradise who will complete his nursing. In a dream the Prophet ﷺ was shown Ibrahim, peace be upon him, in a garden surrounded by children who had died on the fitrah.
A reward for grieving parents: the Prophet ﷺ said that any Muslim who loses three children before puberty will be admitted to Paradise by Allah's mercy to them. When a woman asked about two, he said: and two. Patience over the loss of even one child is greatly rewarded.
Children of non-Muslims: the hadith of the dream mentions "the children of the polytheists" among those with Ibrahim, and many scholars take this to mean they are in Jannah. Others said Allah knows best what they would have done, or that they will be tested on the Day of Judgement. The question is about Allah's justice and mercy, and every view agrees that no child is punished for a sin it did not commit.
The Quran also promises that believing families will be joined together in Jannah. For the practical side, see how to pray janazah and the dua for the deceased, and read about Jannah.
The evidence
Quran, Surah At-Tur 52:21
وَٱلَّذِينَ ءَامَنُوا۟ وَٱتَّبَعَتْهُمْ ذُرِّيَّتُهُم بِإِيمَٰنٍ أَلْحَقْنَا بِهِمْ ذُرِّيَّتَهُمْ وَمَآ أَلَتْنَٰهُم مِّنْ عَمَلِهِم مِّن شَىْءٍۢ ۚ كُلُّ ٱمْرِئٍۭ بِمَا كَسَبَ رَهِينٌۭ ﴿٢١﴾
And those who believed and whose descendants followed them in faith - We will join with them their descendants, and We will not deprive them of anything of their deeds. Every person, for what he earned, is retained.
Believing families are reunited in Paradise.
Sahih al-Bukhari 1248
Sahih... يتوفى له ثلاث لم يبلغوا الحنث إلا أدخله الله الجنة بفضل رحمته إياهم ...
"No Muslim loses three children who had not reached puberty except that Allah admits him to Paradise by His mercy to them."
Children who die young are in Paradise and benefit their parents.
Sahih al-Bukhari 1382
Sahih... إن له مرضعا في الجنة ...
"He has a wet nurse in Paradise."
The Prophet's son Ibrahim, who died as a baby, is in Paradise.
Sahih al-Bukhari 7047
Sahih... وأما الولدان الذين حوله فكل مولود مات على الفطرة ...
"As for the children around him, they are every child who died on the fitrah. Some Muslims asked: and the children of the polytheists? He said: and the children of the polytheists."
Part of the dream in which children who died were with Ibrahim, the basis for the view that they are in Paradise.
Where the ruling is recorded
- al-Minhaj Sharh Sahih Muslim (al-Nawawi), Kitab al-Qadar, chapter on the children of disbelievers and Muslims who die. Calls the view that children of non-Muslims are in Paradise the sound one.
- Fath al-Bari (Ibn Hajar), Kitab al-Jana'iz, chapter on the children of the polytheists. Lists the views of the scholars.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.