No, you can't do that. You'd have to just set it in a separate statement:
var obj = new MyObject();
obj.Test = obj.Id;
The right-hand side of the property in an object initializer is just a normal expression, with no inherent connection to the object being initialized.
If this is something you regularly want to do with one specific type, you could add a method:
public MyObject CopyIdToTest()
{
this.Test = Id;
return this;
}
and then use:
MyObject obj = new MyObject().CopyIdToTest();
or with other properties:
MyObject obj = new MyObject
{
// Set other properties here
}.CopyIdToTest();