I got a B- in my biology course this semester and the day I received this grade I checked my requirements for a specific course I want to take next year and it said the minimum requirement was a B-. However when I checked it again the grade is now a B. I am off by 1 point. Can I ask my professor to change my grade?
As Brian says, professors aren't likely to give you a bonus point just so you can take the next class in the series. After all, there has to be a cut-off somewhere. As an anecdote, I have laughed with multiple graduate students about silly grade-grubbing attempts they're sent.
Most universities let you bypass prerequisites with instructor permission. Since you were so close to the cut-off; and it looks like the cut-off changed in the meanwhile, you might have better luck asking next quarter's professor if you can take her class anyway. In my experience, you might have to register later, which could be difficult if this is a popular course, but I would say you have a better chance with that then getting that 1 point out of the other professor.
You can ask, politely, but don't expect a positive response. "Because I need a higher grade to (get into graduate school, keep my scholarship, take a later course,...)" simply isn't a good reason for a faculty member to change a grade that has been assigned correctly according to the course policies.
You'd be in a very different position if you had become aware of a mistake made by the professor in grading an exam or homework or in computing your overall grade in the course.
As a general rule, it is annoying to instructors to get requests for increases in grades that do not have any substantive argument about the quality of the work attached. Asking for a higher grade on the basis of some desirable consequences that will accrue to you is not a good idea (though some professors might accede to this out of impatience with the matter). If you are going to seek a higher grade, it is appropriate to bring some specific issue in grading to the attention of the instructor (e.g., a question being marked incorrectly or overly harshly), and have a bona fide reason for believing that there are grounds to give you a higher grade.
As other have suggested in their answers, in your case it is better to bypass this completely and just seek a dispensation from the entry requirement for the next course from the instructor of that course. I would suggest that when you go to speak to him/her, you go prepared to give a reason of why the instructor should have confidence that you can handle the course despite you not meeting the required grade (e.g., have you done some extra study on that subject since finishing). If you have a half-decent reason, it is unlikely the instructor is going to quibble over one mark, especially if you are keen enough to come and get special permission to enrol.
Since it's not mention in the answers, but only in the comments:
When you ask for permission to enter the new course despite the grade deficiency, offer to do some extra task or exercise to compensate for the lower grade.
After all, the minimum grades are there for a reason - the professor believes that students that don't have this grade do not have the requisite knowledge to follow his course.
An offer that takes you work but not the professor, shows that you are willing to put in effort to get there. Professors really appreciate that, and it will massively increase your chances of getting in.
Since no one mentioned it, you could take a different tack and try the "grandfathered in" approach. If, when you started the course, the requirement for that course you want to take was a B-, you could humbly approach that professor and/or the registrar and let her/him/them know that you had hoped to take the course and at the time you chose to take the preparatory course the requirement was a B-...and ask if they could show some leeway with the requirement in this type of case.
I don't think the prof or registrar has any written-down stipulation in the school's bylaws to cooperate with this request, but sometimes when a change goes into effect, professors and administrators will be more lenient with those, like you, for whom the change happened while they were preparing to take the class. That's possibly because, unlike merely getting something you want for requesting it, the concept of grandfathering in those whom a prior law applied to exists widely in society.