No, it is not checkmate, because your opponent can capture your queen with either the knight on c6 or the king on e8.
Checkmate occurs when you give a check to your opponent and your opponent cannot either move the king out of check or capture the piece that is giving the check or block the check my interposing another piece between the checking piece and the king.
So, in this case -
[FEN "r1b1kb1r/ppQp1ppp/2n2n2/8/1q6/4P3/P1PP1PPP/RNB1KBNR w KQkq - 0 1"]
1.Qd8+ Kxd8 (1...Nxd8)
There is no checkmate.
Consider this hypothetical position. In this case, it would be a checkmate -
[FEN "r1b1kb1r/1pQp1ppp/pB3n2n2/8/1q6/4P3/P1PP1PPP/RN2KBNR w KQkq - 0 1"]
Here, the enemy king cannot move out of check. The piece that is giving the check (i.e. the queen) cannot be captured by the black king, because it is supported by the bishop on b6. There is also no way to interpose a piece between the queen and the king because they are right next to each other.