A harmless North American snake is called a garter snake, not garden snake. Though, technically there is nothing wrong with using that name if you found it in your garden.
One point is not to confuse a garter snake from North American with one from Africa. The African snake is extremely venomous. They are usually referred to as African garter snakes or aptly venomous garter snakes.
The first use of the term is in the late 18th century, but there is no consensus of where the name garter snake came from. Some believe it is a corruption of the German word for garden which is garten. Another theory is that the name garter snake came from the marks on the back of the snake resembling the garters which men use to hold up their socks. There is little evidence for either theory.
Examples
It turns out that the newts are preyed on by garter snakes that have responded by evolving resistance to the original amounts of poison. [NY Times]
Jennifer fell in love for the same reasons teenage girls everywhere do: He pelted her with water balloons. Once, he caught a garter snake and threw it on her. [Boston Globe]
Rice fields provide suitable habitat features for foraging Giant Garter snakes and expand the potential migration corridor between CBMB and habitats beyond the constraints of the channelized waterways. [Business Wire]
One of its inhabitants is an all-black garter snake that has evolved without the yellow stripes that its cousins elsewhere have. [Twin Cities Pioneer Press]