I have quit my job as I wanted to go back to studying but my circumstances changed and so I've started looking for a full time job. I will have an interview soon so I wanted to give my former manager as a reference. But can she ask details like whether the job is full-time or part-time? If she discovers that it's a full-time position she might think I lied about my reason for leaving. Would she even ask and are hiring managers allowed to give out the details about the job I'm applying for?
Yes, when a reference is called by a hiring manager for an in-depth conversation (as opposed to just confirming employment), it's perfectly fine and normal for the reference to ask a few questions about the job. Good reference checkers will actually explain the role that you're being considered for at the start of such a conversation. They want to know if your reference, which is normally a former manager, thinks you will be a good fit for that role.
Additionally, it's also standard and in my opinion obligatory to give your references a heads-up when you provide their contact details to a hiring manager. You'd tell them that they might receive a call from someone at Company X about Position Y. Not giving them that advance warning is rather discourteous. Former managers can also give you a stronger reference if they get to think about your particular strengths for a certain job in advance instead of being put on the spot.
So your issue is that your former manager will discover that you're looking for full-time employment after initially leaving to resume your studies. Assuming that you had a good relationship with her, you should just explain what happened and that you're now looking for full-time employment. Trying to hide this would be strange and could strain your professional relationship.
Remember: plans and circumstances change. Unless there's some crucial information that you're leaving out, I think it's highly unlikely that your former manager will resent you for changing your plans and resuming your career. If anything I'd imagine that she'd be sorry to hear that you were forced to abandon your plans to finish your studies.
If your former manager was happy with your work, it's possible that she might offer you your old job back. If that doesn't fit with your long-term goals for whatever reason (bad fit, low salary, no prospects for growth, ...), just thank her for the offer and give a suitable excuse. That can be anything: looking for work in a different industry or location, wanting to work with another technology, searching for a new challenge and so on.
But can she ask details like whether the job is full-time or part-time?
A reference check is usually a conversation. During that conversation, either party can ask or say anything they like.
When I have provided references for former co-workers, I end up knowing a lot about their potential new employer. They always tell me the position that will be filled by the applicant.
If she discovers that it's a full-time position she might think I lied about my reason for leaving. Would she even ask and are hiring managers allowed to give out the details about the job I'm applying for?
This leads me to believe that you haven't already asked your former manager that you would be using her as a reference. That is a huge mistake!
Whenever you use someone as a reference - ask their permission first, because you want to make sure you choose people who will give you a great reference.
A surprised reference may not be ready to convey your best side to your new employer. A surprised reference may become angry, think you lied, and give less-than-flattering responses to questions being asked.
You would be far better served to ask your manager ahead of time, prepare her for who will be calling, and try to lay the foundation for a great reference.
Anything else is far too risky.
When asking your former boss for a reference (you were going to ask, weren't you?), let him know that you've had a change of circumstances and that you're actually taking a full-time position. It's not the crime, it's the cover-up that gets you.
It would certainly be a bit odd to question the person calling for a reference, but your boss may ask simply to better understand how to answer questions. I can't imagine any situation where a person would be able to call for a reference, and then NOT be able to say the nature of the position (but maybe that's why I'm not a licensed attorney). If it's divulging personal information to say the position is full-time, certainly it's divulging information to say there's a position in the first place, but obviously that's the reason for the call. But either way, you should be open up front, because the information could always get back to him anyway, even after the interview. It is a small world.