If you literally mean all pieces, the simple answer would be no, there are simply too many free pieces to move that you cannot enforce a real zugzwang scenario there.
Note that, if all pieces are still on the board, it means there hasn't been any promotion yet either. Without capturing, the pawns cannot change their files, and continuing on the same file they'll toss on the opponent's pawn.
The side to move is checkmated in 1 is legitimate, but if you add the zugzwang condition that (let's assume the side to move is not under check to make it a legal scenario) without moving that same player would have survived the position, but only because of the necessity to move, they will get checkmated in 1; then we end up in the same disclaimer above: there are simply too many (16!) pieces around to enforce a full zugzwang.
Actually, if being realistic is not a requirement, you can reach a zugzwang scenario with all pieces on board with the real meaning: if not moving were an option, that would be the best move. With mate in 1, that's pretty much impossible to achieve. Also without being realistic, I don't see the point of setting up a zugzwang position, because the idea of such a position is that things were more or less equal up to now, but the next move will cause damage. Whereas in a scenario like this, the opponent already had many trivial chances to win material advantage. Anyway, for the sake of completeness, I'm posting such a position. Anything that doesn't resemble this will mean that one side has too many free pieces, hence that won't work.
1. b3 a5 2. d3 a4 3. f3 a3 4. h3 g5 5. Nc3 g4 6. Nb1 g3 7. Nc3 e5 8. Nb1 e4 9. Nc3 e3 10. Nb1 h5 11. Nc3 h4 12. Nb1 f5 13. Nc3 f4 14. Nb1 d5 15. Nc3 b5 16. Nb1 c5 17. Nc3 c4 18. Nb1 b4 19. Nc3 d4 20. Nb1 c3