After she has been here for 3 or 4 months and if we happen to fall in love, can we get married and she receive permanent status?

Entry into the U.S. with a visitor's visa requires "nonimmigrant intent," i.e. an intention to stay temporarily to visit and then to return abroad. One may enter the U.S. with nonimmigrant intent, however, and then later change his/her mind about intending to remain permanently. If you are a U.S. citizen and your girlfriend enters the U.S. with a visitor's visa having non-immigrant intent, and then three or four months later the two of you decide to become married, then indeed you should be able to sponsor her in a marriage-based adjustment of status case so that she may become a Lawful Permanent Resident (get a "Green Card"). Of course, there are many additional details that determine eligibility. It would be wise for you and your girlfriend to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.