Yes, the icon library is free for anyone with a BYQ account. The icons themselves come from four open-source packs, and each pack has its own licence.
These names come from the packages BYQ uses. The pack's own licence text is what applies, so read it before you rely on it.
You should see the licence file of that icon's pack on GitHub.
A project for a client is still a project that uses the icon, so the same pack licence applies. Check the licence of the pack you picked, and keep any notice it asks you to keep.
Pick a different pack, or ask the pack's authors. BYQ cannot give legal advice on a licence.
Next: copy or download an icon.