What Adam and Eve believed in were some basic beliefs and moral rules.
There is a verse in the Quran which implies that the first religion was that of Noah:
He has prescribed for you the religion which He had enjoined upon Noah and which We have [also] revealed to you, and which We had enjoined upon Abraham, Moses and Jesus, declaring, ‘Maintain the religion, and do not be divided in it.’ Hard on the polytheists is that to which you summon them. Allah chooses for it whomever He wishes, and He guides to it whomever returns penitently [to Him]. [1]
This verse refers to the fact that the religion of Islam, as the last divine religion, is not different in its principles from previous divine religions. But since human life got more developed and probably some sort of distortion was made to the previous religions there was need to a new religion which abrogated the previous one; till Prophet Muhammad (PBUH) was commissioned to teach people about Islam, the last and most comprehensive divine religion.
The verse also indicates that the prophets who brought a religion from God include: Prophet Muhammad (PBUH), Noah, Abraham, Moses and Jesus. Other prophets promoted these religions and did not bring a new one themselves. For example, David and Solomon promoted the religion brought by Moses.
The first among these prophets who brought a religion was Noah, and prior to him there was not what can be termed as religion. That of course, does not mean that Adam and Eve did not believe in, for example God or the Day of Resurrection. Rather, since life was in its primitive state and the society was not that much developed, and thus, there was no need for a religion (in its technical term and including all its teachings and injunctions), we can conclude that what Adam and Eve believed in were some basic beliefs and moral rules.
Notes:
[1] (42:13) شَرَعَ لَكُم مِّنَ الدِّينِ مَا وَصَّىٰ بِهِ نُوحًا وَالَّذِي أَوْحَيْنَا إِلَيْكَ وَمَا وَصَّيْنَا بِهِ إِبْرَاهِيمَ وَمُوسَىٰ وَعِيسَىٰۖ أَنْ أَقِيمُوا الدِّينَ وَلَا تَتَفَرَّقُوا فِيهِۚكَبُرَ عَلَى الْمُشْرِكِينَ مَا تَدْعُوهُمْ إِلَيْهِۚ اللَّـهُ يَجْتَبِي إِلَيْهِ مَن يَشَاءُ وَيَهْدِي إِلَيْهِ مَن يُنِيبُ