You can check with your health insurer. Simply contacting their customer service department should allow you to determine your health insurance coverage inception date.
Alternatively you should have access to the plan document that may have information on your coverage start and end dates.
Contact your agent/broker or the insurance carrier directly. Your agent/broker normally has a web portal through the insurance carriers, where they can check the status of your application or policy at any time. Your ID card has the member services or customer service number on there, which you can call to obtain status as well, or you likely have access to a member portal through the insurance carrier's website. Please note that if you are late on a payment, or inquiring about a new application, there can be a delay time of processing the payment and/or application, normally a few business days to a week or so. Your agent can be most helpful checking status of an application that is not yet showing in the system.
Alternatively you should have access to the plan document that may have information on your coverage start and end dates.