Did Adam know about the Tree of Life?

Ge 2:

8 The Lord God planted a garden in Eden, in the east, and there he put the man whom he had formed. 9 And out of the ground the Lord God made to spring up every tree that is pleasant to the sight and good for food. The tree of life [T1] was in the midst of the garden, and the tree [T2] of the knowledge of good and evil.

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Both T1 and T2 were centrally located and easily accessible by Adam.

15 The Lord God took the man and put him in the garden of Eden to work it and keep it. 16 And the Lord God commanded the man, saying, “You may surely eat of every tree of the garden,

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including T1, although God did not mention it by name explicitly

17 but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat of it you shall surely die.”

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God warned against eating the fruit from T2.

Ge 3:

22 The Lord God said, “Behold, the man has become like one of us in knowing good and evil. Now, lest he reach out his hand and take also of the tree of life and eat, and live forever—”

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Did Adam know about this special function of T1?

The text did not say. Adam might not know the name of T1 or its special function. As readers, we don't know this special function until this verse.

23 therefore the Lord God sent him out from the garden of Eden to work the ground from which he was taken. 24 He drove out the man, and at the east of the garden of Eden he placed the cherubim and a flaming sword that turned every way to guard the way to the tree of life.

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Adam no longer had access to T1.

Did God tell Adam the name of T1 and its special function?

I tend to be slow to generalize. I don't think God did. This is consistent with God's MO of gradual revelation. Ultimately, Jesus is the tree of life (Jn 14:6, 15:1).