Does crossover happen between chromosomes from grandparents?
Your question and your rephrased question aren't asking the same thing, so you may want to edit your question.

Crossover occurs during meiosis, which is a type of cell division that occurs in the generation of gametes, during which of course only the chromosomes from one of the eventual parents would be present. Meiosis doesn't occur in the fetus, except eventually in the germ line cells.