Yes. Go to HappenBoard → Settings → Migration and pick your old plugin. HappenBoard imports from The Events Calendar, Events Manager, Sugar Calendar, EventKoi and Amelia (its events, not its appointments), and from any .ics calendar file.
It brings over your events with their categories. Venues and sign-ups come from every plugin except Sugar Calendar, and organizers from The Events Calendar. Repeat rules from The Events Calendar, Events Manager and Sugar Calendar are kept when they can be converted. HappenBoard reads the old plugin’s data straight from the database, so that plugin can already be deactivated.
Click Preview (dry run) to see what would be created without changing anything, then Migrate for real. Running it again skips what is already imported.
Migrating from another plugin has the details. For calendar files, see importing an .ics file.