Can I shoot a fish with crossbow?

Yes.  You must have an Idaho fishing license and can only use this type of gear on nongame fish such as carp and suckers.  More information: Archery and Spear Fishing:  Fishing with a bow and arrow, crossbow, spear or mechanical device, excluding firearms. This type of fishing is permitted only in the taking of bullfrogs and unprotected nongame fish and only in those waters during the season set for the taking of game fish.