Full text search SQL server with parameter

You are searching for records which contain the string "@fullname". If you want to search for records which contain the value of the @fullname parameter, you need to concatenate it into the search term:

CREATE OR ALTER PROCEDURE dbo.SelectFullName
(
    @fullname nvarchar(16)
)
As
BEGIN
DECLARE @query nvarchar(20);
    
    SET NOCOUNT ON;
    SET @query = N'"*' + @fullname + N'*"';
    
    SELECT
       *
    FROM
        dbo.NamePersonTB
    WHERE
        CONTAINS (fullname, @query)
    ;
END