My company transferred from NY to NJ mid 2006. I paid NY Sate taxes for 5 months and NJ Sate taxes for 7 months. I'm using Turbo tax and according to the State returns I owe too much. Can I claim back the amount I paid to NY on my NJ return? I do not see anywhere in Turbo Tax where I can do this.

It is likely that TurboTax is combining your total annual income on BOTH returns.

You should be paying NY taxes on the NY-based income and NJ taxes on the NJ-based income.