How to replace a string to an int in JSON value in database column?

You can parse the json with openjson and then rebuild it. This will works, but it is probably very inefficient:

declare @j nvarchar(max) = '[
 {
  "id":1,
  "relatives_birthdays":[ 
  { 
    "month":"8",
    "day": "1",
    "description": "Birthday mother"
  }, 
  { 
    "month":"3",
    "day": "27",
    "description": "Birthday brother"
  }, 
  { 
    "month":"4",
    "day": "12",
    "description": "Birthday father"
  }
  ]
 },
 {
  "id":2,
  "relatives_birthdays":[ 
  { 
    "month":"12",
    "day": "11",
    "description": "Birthday mother"
  }, 
  { 
    "month":"1",
    "day": "2",
    "description": "Birthday brother"
  }, 
  { 
    "month":"7",
    "day": "18",
    "description": "Birthday father"
  }
  ]
 }
]'

select rt.[id]
 , relatives_birthdays.[month]
 , relatives_birthdays.[day]
 , relatives_birthdays.[description]
from (
 select p.id
 from openjson(@j) with (id int) p
 ) as rt
inner join (
 select p.id
  , c.month
  , c.day
  , c.description
 from openjson(@j) with (
   id int
   , relatives_birthdays nvarchar(max) as json
   ) p
 cross apply openjson(relatives_birthdays) with (
   month int
   , day int
   , description nvarchar(max)
   ) c
 ) as relatives_birthdays
 on rt.id = relatives_birthdays.id
for json auto

This is the resulting json:

[
  {
    "id": 1,
    "relatives_birthdays": [
      {
        "month": 8,
        "day": 1,
        "description": "Birthday mother"
      },
      {
        "month": 3,
        "day": 27,
        "description": "Birthday brother"
      },
      {
        "month": 4,
        "day": 12,
        "description": "Birthday father"
      }
    ]
  },
  {
    "id": 2,
    "relatives_birthdays": [
      {
        "month": 12,
        "day": 11,
        "description": "Birthday mother"
      },
      {
        "month": 1,
        "day": 2,
        "description": "Birthday brother"
      },
      {
        "month": 7,
        "day": 18,
        "description": "Birthday father"
      }
    ]
  }
]