You should talk to HR. They know your policy best.
From insure.com:
"You generally start receiving money from your STD policy within one to 14 days after becoming sick or disabled. The actual time for coverage to kick in depends on whether you suffer an illness or injury. If you suffer an injury, your benefits will be paid immediately. If you suffer an illness, it will take longer because there needs to be enough time to show that the illness is grave enough to be disabling."