Can new employer initiate H1b non cap if not travelled on previous H1B

Hey,

My H1B expired this jan. I never travelled using the visa.

 

If I quit, can my next employer initiate non cap H1B?

My current employer has initiated non cap H1B , but its in Intial review status.

Yes, they can file cap-exempt petition for you on the basis of previously approved H-1 petition.