populating int array that is a member variable

You can't use array initializers like that in a class constructor. Members can only be initialized by using the initializer list. Your best bet is to load this data from a file, or to declare the array as static.

class Map
{
private:
static int mapArray[15][20];
/* ... */
};

Then define storage for the static array in a source file (.cpp), also known as a translation unit:

int Map::mapArray[15][20] = { ... };

You should be able to do something like this:

class Map
{
  int mapArray[15][20];

  public:
  Map() : mapArray( (int[15][20]) {
    { 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 },
    { 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39 },
    { 40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59 },
    { 60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79 },
    { 80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99 },
    { 100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119 },
    { 120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139 },
    { 140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159 },
    { 160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179 },
    { 180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199 },
    { 200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219 },
    { 220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239 },
    { 240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259 },
    { 260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279 },
    { 280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299 }
  } )
  {
  }
};

Note that this initialization style is part of C99 and may not be included with your compiler or may not work for C++. Works on GCC 4 for me.

Also note that I did not compile this code. I adapted it for your case from some other code of mine that does work though.

You can use the brace syntax only for initialization.

When your constructor is executed, the array member has alredy been created.