Modify a key-value in a json using jq in-place

Use a temporary file; it's what any program that claims to do in-place editing is doing.

tmp=$(mktemp)
jq '.address = "abcde"' test.json > "$tmp" && mv "$tmp" test.json

If the address isn't hard-coded, pass the correct address via a jq argument:

address=abcde
jq --arg a "$address" '.address = $a' test.json > "$tmp" && mv "$tmp" test.json

AFAIK jq does not support in-place editing, so you must redirect to a temporary file first and then replace your original file with it, or use sponge utility from the moreutils package, like that:

jq '.address = "abcde"' test.json|sponge test.json

There are other techniques to "redirect to the same file", like saving your output in a variable e.t.c. "Unix & Linux StackExchange" is a good place to start, if you want to learn more about this.