Improve randomization function (from 1 to 2)
Randomness

This answer suggests that Random(2) would be okay for most uses. If you want more randomness, you could for example use an implementation of Mersenne Twister. This page lists a few options for Pascal/Delphi, for example this implementation.

Alternating values

Now that I know what you are looking for, I would write the code like this:

// If Value is divisible by 2 then return 1, otherwise return 2
procedure AlternateValue(Value) : ShortInt;
begin
    if Value mod 2 = 0 then Result := 1 else Result := 2
end;

// Your code
randomize();
RandomTurn := random(2)+1;
NumberOfMoves := RandomTurn

while (1) do
begin
    PersonTurn(AlternateValue(NumberOfMoves))
    NumberOfMoves := NumberOfMoves + 1;
end;


You should swap the two lines in the while loop if you want to start with the same number as Random gives. From what I gather, it should not matter much, however.

(I have never written any Pascal or Delphi before, so the code and/or syntax may not be perfect or correct.)