In most cases, when a new license for an app is purchased, the new license key can be retrieved from Atlassian. Replace the trial license key with the purchased one in the Manage apps configuration page.
Log in to your My Atlassian Account where you will see the purchased license. From there, you can find the key:
Go to the Git Integration for Jira app configuration (Jira Administration > Manage apps).
Click the Edit icon to enter/change the License key.