There is no any usage tracking out of box in plugin. There does not seem to exist any way of doing this in Jira. May be plugin could use some online service, like Keen IO, but it would not work in Jira behind firewall, please see issue#1093 also.
However there are some possibilities to see how plugin is used:
Gadgets - it's possible to find if plugin gadgets are installed on any dashboard using SQL query connected to your DB, e.g.:
SELECT portalpage, gadget_xml, count(*) FROM portletconfiguration WHERE gadget_xml LIKE '%jira-timesheet-plugin%' GROUP BY portalpage, gadget_xml;
Reports - it might be possible to see web server access log, and grep for ConfigureReport.jspa.*jira-timesheet-plugin:report
Subscriptions - you can see if there are any Time Sheet Report subscriptions under Administration - Add-Ons - Timesheet Subscriptions
If this is not sufficient, please feel free to add request, and describe your wishes in details, what kind of usage you would like to measure, etc.
For SQL Server:
SELECTportalpage, cast (gadget_xml as varchar(max)), count
WHERE cast (gadget_xml as varchar(max)) LIKE '%jira-timesheet-plugin%' group byportalpage, cast (gadget_xml as varchar(max))