Can girls have a growth spurt at the start of menarche?

Yes, it is completely possible for a girl to have a growth spurt after her first period. The idea that menarche (first period) marks some stop in development is false, menarche is one of many changes that occur during puberty. Puberty continues until adulthood, menarche on average starts at 13 years old whereas puberty tends to end around the age of 18 years old, so a girl may see a growth spurt later in their teens long after they've started to menstruate.