Here is the problem: if it is a STD policy that is partially or fully funded by the employer then you would have to continue to pay the entire policy premium in order to receive the checks. Once the premium is no longer being paid then the checks stop. There is no COBRA continuation for STD policies and it is entirely up to how the policy is written as to whether you can pick those payments up and continue the policy. Often it is written to say that when employment is terminated so does the policy.
If the employer was continuing to pay you out of their generosity then no.
I disagree that you should wait and see if the checks continue. The last thing you want is to be paid money you aren't supposed to get and have them come after you to get it back or worse, find out you missed a deadline or paperwork you need to continue the benefits. You need to contact your benefits administrator ASAP and find out what the policy is on STD and payment after termination.
"That's just my opinion, then again I might be wrong." Dennis Miller