For the month you should use a subquery for get all the month you need:
select monthname(my_date), ifnull(count, 0)
from (
      select str_to_date('2018-01-01', '%Y-%m-%d') as my_date 
      union 
      select str_to_date('2018-10-01', '%Y-%m-%d') 
      union 
      select str_to_date('2018-11-01', '%Y-%m-%d')
      union 
      select str_to_date('2018-12-01', '%Y-%m-%d')
    ) t  
left join  (
SELECT 
    count(v.visit_id) as count,
    MONTHNAME(v.updated) AS Month_name,
    MONTH(v.updated) as my_month
  FROM patient_status as  p 
  INNER JOIN  visit_history_details as v   
    ON v.visit_id = p.visit_id 
  WHERE v.hospital_code = 'id' 
  and p.doctor_id = '2' 
  and v.updated >= now()-interval 3 month 
 
  GROUP by Month_name
) r on r.my_month = t.my_month
And you should avoid old implicit join syntax and use the explicit syntax.
Or as suggested by Salman A for a more general solution:
select monthname(my_date), ifnull(count, 0)
from (
      select LAST_DAY(CURRENT_TIMESTAMP) + INTERVAL 1 DAY - INTERVAL 1 MONTH as my_date 
      union 
      select  LAST_DAY(CURRENT_TIMESTAMP) + INTERVAL 1 DAY - INTERVAL 2 MONTH
      union 
      select  LAST_DAY(CURRENT_TIMESTAMP) + INTERVAL 1 DAY - INTERVAL 3 MONTH
      union 
      select  LAST_DAY(CURRENT_TIMESTAMP) + INTERVAL 1 DAY - INTERVAL 4 MONTH
    ) t  
left join  (
SELECT 
    count(v.visit_id) as count,
    MONTHNAME(v.updated) AS Month_name,
    MONTH(v.updated) as my_month
  FROM patient_status as  p 
  INNER JOIN  visit_history_details as v   
    ON v.visit_id = p.visit_id 
  WHERE v.hospital_code = 'id' 
  and p.doctor_id = '2' 
  and v.updated >= now()-interval 3 month 
  WHERE v.visit_id = p.visit_id and v.hospital_code = 'id' 
      and p.doctor_id = '2' and v.updated >= now()-interval 3 month 
  GROUP by Month_name