As you know North Carolina and Virginia are part of the Drivers License Compact (DLC) and thus if you are convicted of this speeding ticket for 13 mph over the limit in VA then their courts will inform the NC Department of Transportation (DOT).

Once the NC DOT turns this information over to their Division of Motor Vehicles, which keeps drivers' records, it should be placed on your driving record according to the Driver Handbook which talks about out of state convictions and how they are listed on your driving record. As you stated though points are not typically put on your NC driving record for out of state tickets.

Whether your auto insurance rates will go up due to this one moving violation going on your North Carolina driving record will depend upon your insurer's, Geico, rating system. With many insurance carriers a first minor moving violation conviction will not cause your rates to rise but it you can lose your safe driver discount. Since insurance company's rating systems do differ it would be best to call your Geico agent and ask about their specific rating system.

Calling your agent and discussing their rating system and how a speeding ticket, if you are convicted of it, cannot cause your rates to be raised. You may decide to fight the VA ticket or see if there is a way to reduce it to a non-moving violation, etc and thus your insurance provider cannot rate you taking into consideration the speeding offense until you are convicted of it and it goes on your driving record.

If you are convicted of the Virginia speeding violation and it goes on your NC driving record then the next time Geico pulls your MVR they would see it and then it could affect your car insurance premiums. If your rates do rise you can shop around for more affordable car insurance.