I was born in Melbourne but bred in Sydney
No. "Born and bred" is a set phrase, and when used separately its meaning changes. As a standalone word, "bred" is more suitable for use in regards to cattle or other animals. "Born and bred" is tied together so tightly that you can use it as an attributive adjective:
She was a born and bred Melbourner.
What to pick instead?
If by "education" you mean your basic education, including that given by your mother, father or other caregiver, you might say
I was born in Melbourne but brought up in Sydney.
I was born in Melbourne but raised in Sydney.
You can also try
I was born in Melbourne but grew up in Sydney. (Thank you, Kevin, for your comment!)
Instead of "received my education", you can pick shorter versions, such as "(was) educated, tutored" or "graduated", or "went to school/college/university":
I was born in Melbourne but educated in Sydney.
I was born in Melbourne but went to school in Sydney.
P.S. "Born and bred" does remind me of the so-called "legal doublets", but in such combinations the meanings of the words are synonymous, and one of the words usually derives from Latin. I'm not sure that that is the case with "born and bred".
According to Wiktionary, one of the meanigns of breed, most likely an archaic one, is "to take care of in infancy and through childhood; to bring up". Judging by the quotations used for this sense, it must be archaic:
"Ah, wretched me! by fates averse decreed
To bring thee forth with pain, with care to breed"
(by Dryden, a poet who lived in the 17 century)
It's probably due to this "human-related" sense fading away and only the "animal-related" sense retained that we cannot separate born from bred.