My subscription to Norton 360 is good for the next few months. If I subscribe to the new Norton Security now, will it include the months I have remaining within my existing Norton 360 subscription?

It will not.

Active subscriptions cannot be transferred between different products. Therefore, Norton Security and Norton 360 subscriptions cannot be combined, that is, transferred among.

If one opts to buy Norton 360 before the subscription to an older product expires (i.e., found a good Norton 360 deal), that is OK as the usage count for the new subscription starts only after the product is installed for the first time.