Yes! Business Center does this by using Statuses for the client (remember that client could be Job, Property, Account, Pet, depending on your setup). 

Here is an example of a status on the client card, in the top right.

By default, the statuses are Lead, Customer, VIP, and Inactive. However, you can customize these by changing the names of these statuses and/or adding custom ones. The key relationship between them to remember is Lead and Customer, as a lead will automatically become a customer when they A. Pay you or B. Complete an appointment with you. Other statuses you can transition manually

BONUS: This also just got WAY easier to manage in the last update. If you haven't already, try switching your client screen to the Beta view. Then, what you can do here is click on Columns and drag status to the top. That way, on your client screen you can quickly see (and sort) which statuses a client (or Project, Job, etc) is in without having to click all the way into the client to see it. In the next update (which you can expect some time in August) you will also have the ability to filter by a certain status.

Let us know in a comment if this works, if you run into anything, or have feedback on how it can be improved. The new client screen is still in beta so we are actively seeking feedback, thanks for your post!