The detectives have a damn PICTURE OF COLE after George Washington Bridge, 1 hour before the stabbing. So it was midnight, the detectives have proofs that Cole was entering the city that time. There is no way Cole was at Alison's institute!
Also, the detectives told Cole that if he didn't tell them where he was they would consider him a number 1 suspect, NOT THAT THEY WOULD HAVE ARRESTED HIM because they can't arrest him if they don't have real proofs that Cole stabbed Noah, yet. They need to find the knife first. Cole was arrested because of his wrong behaviour with the detectives.
So the whole jail scene makes sense to me. He spent just few hours there because of his behaviour. So he may have not told the police where he was that night and they let him go. He was at the montauk police station. He was not with the detectives who were working on the case! The montauk police was not working on the case, so they could release him.
Maybe he lied to Alison when he told her that he was at her institute because, yes, the institute was in NJ and cole was trying to find an alibi in Alison's presence even if it wasn't true. Alison would always believe him anyway. And when he came back home, he just told Luisa the same things (more or less) he told Alison.
In this case the detectives may still be after him, but they can't arrest him.

I think Cole lied.