Graduating senior: seeking advice

Hi,
I am a senior at a target school graduating next week. I am majoring in STEM and wanted to do something like quantitative risk management.

The recruiting season was VERY tough. I had multiple interviews, and I got one offer at a buyside firm as a quantitative risk position. However, it is a 3 month internship with potential to full-time. I still very appreciate that I got this opportunity, but I am a bit worried that it may not convert to full-time for whatever reasons.

The interview process was very tough. The first round interview consists of doing quant project and writing a report. I spent >50 hours to do analysis and write the report. The second round interview was a phone interview with the hiring manager, mostly technical and questions related to my quant project report.
About 8-10 people got invited to the third round interview (superday) and they said the best candidate will be invited one more time to talk to the founders of the company. I was the best candidate, so I got invited one more time and talked to the founders (4th round) and finally got an offer.

I know that most people in the firm have worked there for a long time, and they don't really fire/layoff employees, and that's why they were very serious about the recruiting process. I am still concerned that the internship may not convert to full-time. What are the best moves I can do for now? Should I talk to the hiring manager to ask for advice? (e.g. asking him about his expectations of me during the internship period, etc).