I think c6... But I don't know if it's right.
Only one winning move

Has to be some sort of corresponding squares problem. I think Kb5 doesn't work because white plays Ke7 and you can't get to d5 because after Kc4 and Ke6 you're not making inroads into black's position. So I'd guess Ka4 (least commital) and a triangulation manouvre (b5 then b4 or just straight to b4) depending on which square black picks. But I'm not very good at this sort of problem.

Are the corresponding squares c4 and e6? If white steps on c4 while black is on e6 he wins. Since if d6 then c6, or if king moves white king gets to d5.

Nice problem!
c4 and e6 are obvious key squares. So, while white stays next to c4 with his king, black needs to stay next to e6. Black can do this from e7, f7 or f6. The White can't advance on the queenside, so the winning plan seems to be grabbing the pawn on g3 via Kd3-e2-f3.
If blacks king moves to f6 after we move to e3, he can defend the pawn with Kf6-g5-h4 and we can't progress. So we need to reach d3 with the black King on f6 so he must move away.
So with this in mind, let's see:
1. Kb4? Ke7! doesn't win, because white can choose to loose a tempo. For example:
2. Kc3 Kf7!
3. Kd3 Kf6! with a draw (see the thoughts above. Also notice, that white can't win a tempo with a triangle-maneuver Kc3-b3-b4-c3, because black also has such a triangle available).
or 2. Kb3 Kf6! 3. Kc3 Kf7 4. Kd3 Kf6! with a draw.
1. Kb5 gives similar lines. Black just moves from e7 to f7 and back until white plays Kd3.
So our only winning chance seems to be 1.Ka4
1.Ka4! Ke7? loses easily:
2.Kb4! and now either 2. .. Kf7 3.Kc3! Ke7 (Kf6 fails to 4. Kd3) 4. Kd2! Kf7 (Again Kf6 fails to Kd3) 5. Ke2! Ke7 (you guess, Kf6 fails to Kd3) and 6. Kf3! and white is winning.
1.Ka4! Kf7 seems a bit more tricky. Now, 2. Kb4 Ke7 3. Kc3 Kf7 4. Kd2 doesn't work due to 4. .. Kg6 (note how that move wasn't available in the line with 1. .. Ke7). Neither does 2. Kb5 Kf6 3. Kb4 Ke7 or 2. Kb3 Kf6 3. Kc3 Kf7.
The only move left is 2. Ka3!
2. .. Ke7 3. Kb4! and white wins.
2. .. Kf6 3. Kb3! Kf7 (3. .. Ke7 4. Kb4! wins) 4. Kc3! Ke7 (4. .. Kf6 5. Kd3 wins) 5. Kd2! and white wins as above.
2. .. Ke6 3. Kb2! Ke7 (3 .. Kf7 4. Kc3! wins as well as 3. .. Kf6 4. Kb3! Kf7 5. Kc3! or 4. .. Ke7 5. Kb4!) 4. Kc2!
and now every move seems to lose, for example 4. .. Kf7 5. Kc3!, 4. .. Kf6 5. Kd3!, 4. .. Ke6 5. Kd2!
wow, now that got much longer as I expected. Hope I didn't miss anything important...
Which one?