Change Security Group

No.  The security group or groups are selected at the time of creation.  Once the VM is created, you cannot change or add new security groups.  If you do not select one of your own making, a default security group will be associated with the instance.

You can modify the default group--or any other security group you assigned at creation--at any time so you may be able to avoid having to recreate your instance.  At worst, you can take a snapshot of the existing VM, create a new template from it and create a new VM from that, defining additional security groups at that time.

CAVEAT: Changes you make to a security group go into effect immediately, and they effect every VM with which that security group is associated.