MySQL 5.7

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CREATE TABLE contacts_table( first_name varchar(50),  last_name varchar(50), full_name varchar(100) )
 
INSERT INTO contacts_table(first_name ,  last_name)
VALUES('John','Doe'), ('Peter',null), (null,'Brown')

Records: 3  Duplicates: 0  Warnings: 0

UPDATE contacts_table
   SET full_name = TRIM(CONCAT( COALESCE(first_name,''), ' ', COALESCE(last_name,'') )) 
   
Rows matched: 3  Changed: 3  Warnings: 0

SELECT * FROM contacts_table

first_name
last_name
full_name

John
Doe
John Doe

Peter
null
Peter

null
Brown
Brown