Similar methods using loops
Not so similar...

Just because the method bodies "look similar" doesn't make them similar. It's better to think in terms of purpose instead. The two methods are for completely different things. They are not similar, and good to be separated.

In functional languages you could use a "walk" method to iterate over all cells of the board and use a callback function to act on each cell. You could emulate that behavior in Java, but the code will get more complex, and I'm not sure it will be worth it in this example.

Robustness

Do you intend to initialize the board more than once in the lifetime of the object? If not (probably), then move the initialization to the constructor.

Writing style

You should use camelCase for naming, for example initializeBoard instead of initialize_board.



It's recommended to use braces with ifs, for example:

if (x == 0) {
    board[x][y] = (char)y;
} else {
    board[x][y] = '~';
}




It's better to print a newline using println:

System.out.println();
// instead of: System.out.print("\n");