$refs are only populated after the component has been rendered, and it is not reactive. It is only meant as an escape hatch for direct child manipulation - you should avoid using $refs in templates or computed properties.
This lets me dynamically set the ref for a Vue component. Once you have a ref you gain access to the Vue components instance (vars, methods). Classes are dynamically manipulated differently in Vue. See the reference guide.
$refs point to Vue components. clockpicker is dependent on JQuery. If you want to port it you have to make it an actual Vue component. Or just use JQuery in your Vue app, its easily done. But JQuery components will not show up in your $refs array unless it is wrapped into Vue component.