There is precisely one, it is called Fool’s mate, and Black gives it: 1.f3 e5 2.g4 Qh4 mate.
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e f g h 1 White cannot do it. Not against best play, not against terrible play, not at all. After two white moves and one black move there is simply no legal arrangement of pieces where the black king is attacked with no escape. The white queen would have to get to a mating square and the black king would have to be exposed, and one black move is not enough to expose him. The best White can manage is mate on move three, which is a separate story and still needs Black to help twice.
There is a whole page on the two-move mate itself if you want to see why those two pawn moves are so bad.
What a puzzle means by mate in 2
Different thing entirely. When a puzzle says mate in 2, it means two of your moves. Your opponent replies in between, and here is the part that catches people: your first move has to work against every legal reply, not just the one you were imagining.
That is why a two-mover is so much harder than a one-mover. In a mate in 1 you look for a check that ends it. In a mate in 2 you usually look for a quiet move, a move that gives no check at all and simply takes away every square the king has left.
8 c g Count the checks in that position: 1.Rb8+, 1.Rc8+ and 1.Rh2+. Every one of them lets the king out, because after 1.Rb8+ Kh7 the rook on c1 needs another move to catch up, and by then the king is on h6.
The solution is 1.Rb7, and it is mate in two because it is not a check. The rook takes g7 and h7 away, the black king has exactly one legal move in the whole position, 1…Kg8, and 2.Rc8 is mate. Two rooks working like this is a pattern of its own, the ladder mate, and it is the first mate most beginners learn to force.
How to solve one
Work backwards from the mating picture rather than forwards from the checks. Ask what the final position would have to look like: which piece gives the mate, from which square, and which of your pieces covers the flight squares. Then ask what is missing, and whether one move puts it there.
If the king still has two escape squares after your best check, the check is not the answer. Look for the quiet move instead. That single habit is most of what separates people who solve two-move puzzles quickly from people who stare at them, and it is the same skill composed problems train, which is why chess problems are worth an occasional hour even if you never play a tournament.