jq: error: test1/0 is not defined at <top-level>, line 1

Using shell-based variable interpolation is usually a very bad idea. In the present case, what if the shell variable contains a double-quotation mark, for example?

The safe thing to do is to use env or --arg or --argjson as appropriate, e.g.

jq --arg e "${Environment_Name}" '.environments[$e] += ["test1"]'

You have some extra quotes in there and test1 needs to be ["test1"]

jq ".environments.${Environment_Name} += [\"test1\"]" tmp.json