You certainly can negotiate a fixed cost contract. Be aware, however, that this sort of thing is quite risky particularly when you don't have a lot of consulting experience. I would be very, very cautious about doing so for anything close to an internship.
You bear the risk of misestimating. Most developers, particularly younger developers, suck at estimates. Perhaps your estimate of a month is right on the money. Perhaps their 3 month estimate is closer to reality. What happens, though, if the project takes 4 or 5 months? Almost everyone that does fixed bid contracts has a horror story (or three) about a project that they lost their shirt on. If you aren't highly familiar with their existing code base, a project to integrate your code with their existing application is quite risky-- you never know what struggles you're going to have because of issues in their code that you're not aware of.
You bear the risk of scope creep. Documenting the complete scope of a project is a skill that, again, most developers are not good at. In the real world, that means that there will frequently be cases where the company asks you to do more work because they're making a very expansive reading of anything that is potentially ambiguous. You, on the other hand, are going to have the financial incentive to make the narrowest possible interpretation of the scope. If you're not comfortable pushing back professionally on requests, you can find yourself getting taken advantage of.
You're taking on billing and tax risks. An intern would normally be a short-term employee of the company. The company would give you a regular check, it would withhold taxes, etc. A fixed price contract, though, means that you'd be a company that would need to invoice them, wait for them to pay (it would be normal that the company would have a month to pay after they received your invoice), follow up when they haven't paid on time, etc. Being a company also has a lot of tax consequences-- you'd owe self-employment taxes, your taxes get a lot more complicated to file, etc.
I'd strongly suggest at that you just try to work as an intern. If you manage to complete the entire project and every bell and whistle that the company can think of in ` month rather than 3, it is unlikely that they are going to run out of things for you to do. Companies always have more work than they have developers to do it. If you exceed their expectations, they're going to find more work for you to do.