It depends on the context. The second sentence is only correct if you are talking about the past of the past. So, if your sentence is already in the past, you would describe the past further back in time with 'had'. My Mom used to cook pasta. She had grown up in Italy, where pasta was hand-made, and cooked every day. Where did your Mom grow up? She grew up in Italy.

Both are grammatically correct, but the first sentence may be used in either the present or past tense. The second sentence also sounds slightly less natural.

My mother grew up in Italy is more grammatically correct and natural.

The first one

My mother grew up in Italy is correct