The Pritchard-Beasley Classified
Encyclopedia of Chess Variants gives a fool's mate in 4 for #HexagonalChess:
1 Qc8 Qc14
2 b7 b15
3 Bb5 e12
4 Qxf17.
They have too much focus on how square chess uses the queen for fool's mate. In hexagonal #chess, opening with the bishops is the nuclear option, especially if Black is thinking in terms of square chess openings with pawns:
1 Bd9 b13
2 Bh5 h11
3 Bj11 mate
Gliński chess starts in the middle game with diagonals open.
Since Gliński #chess starts in the middle game with diagonals open, the worst response in #HexagonalChess to a bishop opening is pawns. That's square chess thinking.
Black's far better(!) options are to (a) open up another place for xyr king to escape to by moving the front bishop out of the way (e.g. respond with the very same fool's mate), (b) get xyr queen's knight across the board early so that it can sacrifice for White's bishop, or (c) do the same sacrifice but with the queen.
Of course Gliński #chess has the mirror image fool's mate available for Black; so if White doesn't go for the bishop-led nuclear option, Black can.
The thing that makes bishops such vital parts of the opening game in #HexagonalChess, and forgetting about them being square-chess thinking, is that Black's mirror fool's mate cannot be countered by White's king's knight, but instead White has either to get xyr king's knight's pawn forward or to counter with xyr own diagonal pieces.
JdeBP