You'd need root access.
/data/data/com.opera.browser/app_opera/
Speed Dial is favorites.db

Would it be accessible via adb on a Nexus device?

So I just got Android 5 on my Nexus 10 and discovered that Opera will only run for 5-10 seconds and then crash.
The beta version seems to work fine but my tabs are on the standard version.
I did notice the beta had an optimizing notification, perhaps that made the difference?

Only if you have root access, e.g. https://forums.opera.com/topic/2139/how-can-i-backup-or-export-my-speed-dials-from-opera-on-my-android-tablet
Oh, please use a better title next time?

Will the built in Google backup save the tabs or speeddials?

Is there a way to backup my Opera info on android?