Yes, it works with cantrips
There is no qualifier for the Stone Sorcerer's Stone's Edge so it applies to all spells, both cantrips and level 1+.
Stacks with Booming Blade
The only limits you have from Stone's Edge are:
When you cast a spell that deals damage, choose one creature damaged by that spell on the round you cast it. That creature
takes extra force damage equal to half your sorcerer level. This feature can be used only once per casting of a spell.
In this case, you have cast two spells that affect the same creature. You can apply the Stone's Edge to both.
But something is weird
It's unclear what "can be used only once per casting of a spell" means. It's likely that this is protect against Quickening a spell to get it twice, but it's something to consider for other combos you may have in mind.