Quad Core or Intel i3 4030-U

That question can't be answered. "Quad core" is too generic. You might be talking about an i7 or an Atom CPU. Clearly identify what things you are trying to compare. Since the i3 is a "U" my guess you are trying to compare laptops.
 
I am going to take a stab and say the quad core you are referring to is the A8-6410 APU with the integrated Radeon R5 (Beema) graphics core. In terms of overall CPU performance the Core i3-4030u is at least just as powerful as the A8-6410. The Intel HD 4400 integrated graphics core is overall a bit more powerful than the Radeon R5 (Beema) graphics core.