SQL Server 2014

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CREATE TABLE DataTableOne (
    ID int,
    DATA varchar(MAX)
);

INSERT INTO DataTableOne (ID, DATA)
VALUES (1, '{"name":"Cole", "update_date":"2855290000"}'),
(2, '{"name":"Peter", "update_date":"1222290000"}') ;
2 rows affected

SELECT
    DTO.*
FROM dbo.DataTableOne AS DTO
CROSS APPLY 
(
    SELECT 
        TRY_CONVERT(xml,
            REPLACE(
                REPLACE(
                    REPLACE(
                        REPLACE(DTO.[DATA], 
                            '"name":', 'name='),
                        ', "update_date":', ' update_date='),
                    '{', '<r '),
                '}', '/>'))
) AS X (x)
WHERE
    1 = X.x.exist('r[1][@update_date ge 1645290000]');


ID
DATA


1
{"name":"Cole", "update_date":"2855290000"}


SELECT * 
FROM dbo.DataTableOne AS DTO
WHERE 
    DTO.[DATA] COLLATE Latin1_General_BIN2 
        LIKE '%"update_date":"164529[0-9][0-9][0-9][0-9]"%'
    OR DTO.[DATA] COLLATE Latin1_General_BIN2 
        LIKE '%"update_date":"1645[3-9][0-9][0-9][0-9][0-9][0-9]"%'
    OR DTO.[DATA] COLLATE Latin1_General_BIN2 
        LIKE '%"update_date":"164[6-9][0-9][0-9][0-9][0-9][0-9][0-9]"%'
    OR DTO.[DATA] COLLATE Latin1_General_BIN2 
        LIKE '%"update_date":"16[5-9][0-9][0-9][0-9][0-9][0-9][0-9][0-9]"%'
    OR DTO.[DATA] COLLATE Latin1_General_BIN2 
        LIKE '%"update_date":"1[7-9][0-9][0-9][0-9][0-9][0-9][0-9][0-9][0-9]"%'
    OR DTO.[DATA] COLLATE Latin1_General_BIN2 
        LIKE '%"update_date":"[2-9][0-9][0-9][0-9][0-9][0-9][0-9][0-9][0-9][0-9]"%';


ID
DATA


1
{"name":"Cole", "update_date":"2855290000"}