Concat 2 fields in JSON using jq

Use parentheses around the string concatenation code:

echo '{"channel": "youtube", "profile_type": "video", "member_key": "hello"}' \
 | jq '{channel: (.profile_type + "." + .channel)}'

Here is a solution that uses string interpolation as Jeff suggested:

{channel: "\(.profile_type).\(.member_key)"}

e.g.

$ jq '{channel: "\(.profile_type).\(.member_key)"}' <<EOF
> {"channel": "youtube", "profile_type": "video", "member_key": "hello"}
> EOF
{
  "channel": "video.hello"
}

String interpolation works with the \(foo) syntax (which is similar to a shell $(foo) call).
See the official JQ manual.