My LG G3 stopped working and went blue screen. I got aN LG G4 and I have no idea how to get my contacts, apps, music, and memory back through my Google account.
Log in your Google account and the contacts should be synced to your phone.
Google Play Music should be the same, but I'm not sure cause I don't use it.
For apps, you have to install them again through Google Play Store. What do you mean "memory"? Usually, Google back up contacts, calendar, some app data, etc.
For contacts, you can check the status on Gmail. Log in Gmail on a PC browser, click the triangle icon next to Gmail (on the top-left), then choose Contacts. If you can find contacts there, then you can restore them to your new phone. I don't have a G4, but you may check the settings option of Contacts app or the backup and restore function.
As far as I know, Google doesn't back up download files unless you uploaded them to Google Drive. You can log in Google Drive and check.