Would the flu shot cause an increase in white blood count?

Stress response: Anything that your body decides is a stress agent can produce a transient increase in your white cell total count. A flu shot would cause a mild elevation in some & none in others.

Possibly: If the flu shot caused some degree of inflammation at the injection site the WBC count may go slightly up. You do not say what the count was and is and what the differential WBC is. Absent other symptoms would not necessarily about it.