Let's not overanalyze this--the answer is no. Garter, Ribbon and Water snakes all eat frogs, as do Ringnecks and, of course, Hognose snakes. King snakes eat anything, and baby rat snakes love frogs.
Want a snake for a frog enclosure? Hmmn--I can't think of a better, easier one than a rough green snake. (Although I have seen zoo exhibits with arboreal boids and large tree frogs, and crotalids with large toads.)
Rough green snakes are insectivorous, and can be kept successfully with anoles and/or tree frogs in big, naturalistic setups. A bit delicate though, so I wouldn't handle one as readily as say, a garter or corn snake.