No, losing a confidence vote in parliament (as opposed to in the Conservative party) would mean dissolving the parliament, not just Johnson stepping down, so the Tories would be unlikely to vote no confidence. However, it is still a threat that Starmer can use, since it would be at least inconvenient.
I understand that, but think the Tories are in such dire straits and don’t look like improving any time soon, that enough Tory MPs could vote no confidence just to rid themselves of the Boris anchor. As a number here have observed his resignation doesn’t mean he’s gone.