Is there a minimum amount of time that I must live in Florida before I can file for an injunction?
No, there is no minimum requirement of residency for you to meet before you file a petition for an injunction for protection against stalking in Florida. You also have the option of filing the petition in the county where the respondent lives, or where the stalking occurred instead of in the county where you currently live.1 This might be the best option for you if you have moved to a new county and the stalker does not know what county you are living in.
1 Fla. Stat. § 784.0485(1)(f)