My resident state is TX, did a remote work to CT employer, do I have to pay state income tax in CT?

No. You are not required to pay CT state income tax.

Thank you so much.  But my W2 withheld CT income tax?  how do I request the refund?  TurboTax is not giving me an option to request refund

Since you never lived in or physically worked in CT, you don't owe income tax to CT. You have to file a non-resident CT tax return (showing zero CT income) only if you want a refund of the incorrectly withheld CT taxes.

Great, really appreciate your response.  Refund worked in TurboTax