You can pass the OR statement between the condition followed with the "statement1" OR "statement2 and make sure you have space between statement1<space>OR<space>statement2"
For my case below is the sample that worked.
https://graph.microsoft.com/beta/users?$search=%22jobTitle:u%22%20OR%20%22displayName:u%22
For your case below should work
https://graph.microsoft.com/v1.0/me/messages?$search="hello" OR "good day"
Note : OR should be in capital.
Thanks