No, the battery voltage does not appear on the J1772 connector, or the equivalent AC charging connectors around the world. It is only on DC charging connectors, whether they are separate from the AC charging connector (such as CHAdeMO or the Chinese system) or they are combined with the AC connector such as in CCS and the Tesla designs.
Even in systems with a DC charging connection, those contacts are not live until the onboard equipment and the charging station equipment (electric vehicle service equipment or EVSE) communicate and negotiate. The AC contacts are not live, either, and when everything is turned on they are normally only the input to the onboard charger, which will not run in reverse to convert high voltage DC from the battery into AC power.
There are power export standards supported by some connections and some equipment, but they usually export only AC power since it is rare for any EV to be connected to a DC charging station in a situation where export of power from the vehicle would be desired. So yes, you can do this in a Type 1 (120 V AC) or Type 2 (240 V AC) connection, but in an OEM EV if the manufacturer didn't design this in (and essentially no one has included it until very recently), it would be a major project.
What you're looking for ideally is
Vehicle to Grid (V2G), or the Vehicle to home (V2H) version of that.
If modifying an existing vehicle, it would likely be easier to make an entirely separate connection of the HV DC from the pack to an inverter, although ~360V DC to 120 V or 240 V AC is not a common combination so off-the-shelf inverter availability would be poor. One variant of the Ford F-150 hybrid is now or will soon be available with this feature, so perhaps the hardware could be salvaged.