Since you are doing a paid internship, you are an employee of the company. If it has a research lab, it probably has a policy on publications and might have a policy on outside work. Ask your supervisor at the company about this situation.
Where I work, the relevant policies are:
- Any paper I publish using my workplace as my affiliation must be approved
for publication by my boss and uploaded to our manuscript management
system before I submit it.
- Any outside work that I do -- even in my own time -- must be approved by my boss in advance. This is particularly important for research work.
The upshot is that if I want to work on and publish a paper while working here, my boss needs to approve it one way or the other (either as outside work without my corporate affiliation or as a publication with my corporate affiliation).