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I successfully passed the the two first steps of the hiring process: a test and an interview.

For the final step of the hiring process they are requiring me to provide my current superior's contact information to complete the process.

I don't want to risk my current job. However, I feel strongly that working for the government would be good for my career. Also, the hiring process with the government has already taken too long (in my opinion) and I don't want to give to give it up.

My immediate superior at my current work is a nice person, so I don't think she would want to give a bad reference. However, this will for sure put her in a delicate position with the higher-ups.

What exactly should I do? Should I ask my immediate superior for her contact info and tell her to expect a call from the government? Should I even allow her to prepare mentally for the call?

p.s. They specifically asked for a reference from my "current immediate superior", but even if I wanted to, I never had another "immediate superior", because my current job is my first "real job".

p.s. I'm located in Canada.

If you want the government job then you will need the reference. Yes, it is a risk for your current job but only you can determine if it is worth the risk. You can approach your supervisor with something like this:

Hey Supervisor, I have been presented with what I feel is a great opportunity with the government and I would like your permission to pass along your contact information and have you as a reference.

If she is a nice person like you mentioned, she will happy for you and gladly provide herself as a reference. She will understand that not everyone stays at their first job forever, and being a supervisor she has likely already seen employees come and go.

I think one should not hesitate to take risk for better opportunities in career. It is mandatory requirement for gradual success and achievement in life that one should clearly set his target and than boldly proceed towards it's achievement. Yes in some cases there may be some failures but without taking risk one cannot proceed quickly in life and in some cases lost better appourtunities.

In your case, luckily you have already cleared written exams, moreover , you are confident that due to your performance in current organization your boss will rate you satisfactorily.
In my opinion in such conducive envoirment it is not risk at all that you should state way tell your boss regarding new appointment in detail with all benefit you can get through new job. This process will convince her to help you in your last formality for new job. A helpful boss will never create hinderess in your bright career

The other answers are good, but I'd like to present an opposite point of view for consideration.

I don't want to risk my current job

That's important. You don't know for sure how your boss will react, or how her boss will react. It's possible that your boss is a kind person who would give you a great reference, but her boss may demand you be fired immediately. You don't know what will happen.

There are plenty of questions here around "when should I inform my employer I'm likely to leave?" (Here's one)

By talking to your current boss, you risk losing the job you have with no guarantee that you'll actually get the job you're applying for. That's a big risk

My recommendation is that you talk to someone in the new organization and see if they can possibly apply some common sense to this situation. They may be able to find a way to check that box without asking you to risk your livelihood.

If you comply with their request and talk to your current supervisor, the worst that can happen is that you lose both your current job and the new job. Are you OK with that?