We have install trial versions of Jira Software in our company and now we purchase the commercial license. Is it possible to make the conversion without reinstalling Jira?
Yes, very much so.
The limitations on a trial Atlassian system are really just "out of time". There is no difference in functionality or how it workds, it's just "trial = ends at some point"
If you like what you have now, just go for it, the default behaviour is actually "these people have used it for a trial, they chose to carry on using it, let's assume the data the did during the trial is valid and not just a test"
You don't need to (re)install anything, just confirm that you are happy to take up usage!