How do I make fill with color only one column of datagrid WPF?

Your Rectangle has nothing to use for a "size request" ... at a minimum, give the Rectangle a .MinHeight and a .MinWidth in this case (e.g. 24x24 or 32x32). If the Rectangle was "contained", and "stretched", you wouldn't need to specify dimensions, but there are no other controls in the template.

(I say .Min... because some controls have a default .Min..., and anything less, won't work if specified via .Width or .Height only).

You do not need a Rectangle to set the background color of a cell, you just need to set the Cell Style. Here is a working example:

<Window x:Class="WpfDataGridCellBackground.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        Title="MainWindow" Height="450" Width="800">
    <Window.Resources>
        <SolidColorBrush x:Key="clBr" Color="Red" />
    </Window.Resources>
    <Grid>
        <DataGrid ItemsSource="{Binding Path=.}"
                  AutoGenerateColumns="False">
            <DataGrid.Columns>
                <DataGridTextColumn Header="First Name" 
                                    Binding="{Binding Path=FirstName}">
                    <DataGridTextColumn.CellStyle>
                        <Style TargetType="DataGridCell">
                            <Setter Property="Background" 
                                    Value="{StaticResource clBr}" />
                        </Style>
                    </DataGridTextColumn.CellStyle>
                </DataGridTextColumn>

                <DataGridTextColumn Header="Last Name" 
                                    Binding="{Binding Path=LastName}">

                </DataGridTextColumn>
            </DataGrid.Columns>
        </DataGrid> 
    </Grid>
</Window>

And the code-behind:

using System.Windows;

namespace WpfDataGridCellBackground;

public partial class MainWindow : Window
{
    public MainWindow()
    {
        InitializeComponent();
        DataContext = new List<Person>()
        {
            new("Freddie", "Mercury"),
            new("John", "Lennon")
        };
    }
}

public record Person(string FirstName, string LastName);