The FMX framework offers a couple of means to change the appearance of the background of a TGrid. Two means, alternating row colors and color per cell are presented in the following.
Alternating row colors, optionally using styles
This exists as a presetable boolean item in the TGrid.Options property named AlternateRowBackground. The default color is a light gray color ($FFEEEEEE). To change this color you can add a TStyleBook or right click the grid and select Edit Custom Style ... or Edit Default Style ... and then change the Color property of gridstyle - alternatingrowbackground. Here an example where the color is changed to Bisque:

Code in the OnDrawColumnCell event
This even is called for each cell of the grid and offers full control of painting the cell background. The header of the event handler looks like this:
procedure TForm11.Grid1DrawColumnCell(Sender: TObject; const Canvas: TCanvas;
  const Column: TColumn; const Bounds: TRectF; const Row: Integer;
  const Value: TValue; const State: TGridDrawStates);
For painting we will need a TBrush, so we declare one as a local variable:
var
  bgBrush: TBrush;
We are now ready to apply some scenarios of special background drawing. First is how to let the default drawing take place for certain cell states.
  if (TGridDrawState.Selected in State) or
      (TGridDrawState.Focused in State) then
  begin
    Grid1.DefaultDrawColumnCell(Canvas, Column, Bounds, Row, Value, State);
    Exit;
  end;
For the following we will need the TBrush, so we create it and manage its lifetime (on Windows platform):
  bgBrush:= TBrush.Create(TBrushKind.Solid, TAlphaColors.White); // default white color
  try
  //
  // following code snippets go in here
  //
  finally
    bgBrush.Free;
  end;
Next, an example of painting alternating row backgrounds without using styles
  if Odd(Row) then
    bgBrush.Color := TAlphaColors.MoneyGreen+$202020; // a very light green color
  Canvas.FillRect(Bounds, 0, 0, [], 1, bgBrush);
Then an example of background color for given columns
  case Column.Index of
    0: bgBrush.Color := TAlphaColors.lightBlue;
    1: bgBrush.Color := TAlphaColors.MoneyGreen;
  end;
  Canvas.FillRect(Bounds, 0, 0, [], 1, bgBrush);
And finally an example of background color determined by value of data
  if Column.Index = 1 then
    if Value.AsOrdinal < 0 then  // negative 
      bgBrush.Color := TAlphaColors.Lightpink
    else
      bgBrush.Color := TAlphaColors.MoneyGreen;
  Canvas.FillRect(Bounds, 0, 0, [], 1, bgBrush);
And a sample image:

The texts are colored as in this answer