Diagonal Movement Code for Python Chess Engine
Go through the PEP-8 style guide. You have inconsistent naming
  convention. A mixed case of camelCase and snake_case throws off
  the developer. Follow snake_case for variables and camelCase for
  class etc.


The first thing that comes to mind is a loop:

for pos in range(1, 8):
    try:
        temp = chessBoard[i + pos][j + pos]
        solutionMoves.append([i + pos, j + pos])
    except:
        break


which about covers the whole of your try-except blocks.



However, with chess; I'd suggest using coordinate system to move around the board.

Design a class Point which takes \$ (x, y) \$ position and define the __add__, __sub__, __neg__ etc. as follows (rough, modify as per your needs):

class Point(tuple):
    def __add__(self, other):
        return Point(v + w for v, w in zip(self, other))

    def __radd__(self, other):
        return Point(w + v for v, w in zip(self, other))

    def __sub__(self, other):
        return Point(v - w for v, w in zip(self, other))

    def __neg__(self):
        return -1 * self

    def __mul__(self, s):
        return Vector(v * s for v in self)

    def __rmul__(self, s):
        return Vector(v * s for v in self)


Now, define your movement direction as a point vector:

DIR_TOP_RIGHT = Point(1, 1)


when you want to move inside the board, just add a multiple of direction to the current point:

current = Point(i, j)
new = current + (distance * DIR_TOP_RIGHT)


Next, define a method (inside a class ChessBoard which checks whether a point lies inside the board or not. This should be quite easy. This ChessBoard class should also be the one responsible for converting your Point object from \$ (1, 1) \$ format to B1 etc.