Yes, state governors usually have bodyguards, generally state police officers. However, given the expense of assigning these officers full-time to protective detail, these expenses often come under scrutiny and criticism by the political opponents of governors. (Plus, the Secret Service only protects the immediate family of the President - spouses and children. Anyone outside of that would not qualify.)
I'm going to say that in this case it depends on your authorial discretion. If the governor is a big-spender, or values the peace of mind over the political heat, he will have security for his daughter. (If the school is out-of-state, they might hire private security or off-duty local police instead to avoid jurisdictional issues of having one state's police operating in another state on a permanent basis.) But if the governor is a penny-pincher, they may forgo any additional security unless a credible threat is made.
does a governor have a bodyguard? What about a lieutenant governor?
Now about the daughter of a governor, I have a similar situation. Would there be extra security at her school (not following her around, just at the gates and things?) Her school is a private elite school so there are other daughters of rich men there (her grandfather was a mayor before, and the family is wealthy as it is)
I did research for a similar-sounding school environment I wrote a few years ago--looked into a couple of real schools in DC that enroll a lot of students of important political figures (domestic and international). One of the reasons parents would choose those schools is BECAUSE of the additional security/safety. It wouldn't be advertised on the school's website, or necessarily visible (ie, no buff men in dark suits with sunglasses and earpieces following everyone around), but the school would know exactly who was supposed to be there and who wasn't, and anyone who wasn't would immediately be questioned and removed, if necessary. Keeping the kids safe is the priority, obviously, but they also want them to have an undisrupted learning environment--especially considering that the majority of the students would be from wealthy families, but under no imminent danger of being kidnapped, etc.