Plenty of booms out there, no need to buy a new stand w/kit.
I'm using a 51cm dual flash boom with the 120cm Godox and, once you add the length of the stand swivel (a necessary addition in order to work at all), the length is just about the minimum you'll need for the Godox. Also note my previous post showing my tests with the 120cm Godox, it really wants/needs dual flashes to get full fill so I would certainly consider buying a dual flash boom so as to have the option of adding that additional flash later on.
The secret is the stand adapter, the "Swivel Umbrella Adapter", it is the object that will take all the weight once you tilt the setup. I recommended looking for ones that have a very deep upper adapter section, like this
not like this
It is difficult to tell but note how deep the top spigot section is on the first one, versus the shallow upper spigot fitment on the second. The deeper upper spigot holds the boom arm more securely under a heavy load. It is even better if the swivel adapter has a notched joint, that is the adjustments operate in steps due to the joint being ratcheted, rather than infinite adjustment with a smooth joint. That seems counter-intuitive, it means less fine adjustment, but the ratcheted joint easily stays locked solid under a heavy burden versus the infinite, smooth joint which can slip.
And, trust me, if you do indeed go with the dual flash setup you'll need that strength: add the softbox weight along with its interaction with the environment (acting like a sail), the dual flashes, the 8 AA's for the flashes, the 2 triggers, the batteries for the 2 triggers plus the weight of the boom arm itself...it gets heavy!
The stand that you have looks very similar to my own heavy-duty stands, so it will hold the weight. It's up to you to make sure that you do the boom arm right so that it doesn't fall on anyone's head!