Depends on your definition of testing, because some UX practitioners would argue that even gathering analytics and using it to guide some of the design decisions is in a way doing user testing, so what you are talking about is probably doing testing without actual users being directly involved in the process.
The two things straight off the top of my head are: analytics and guerilla testing.
With analytics, the aim is to spot patterns and behaviours through a large volume of data that are indicators of underlying user interaction (or micro-interaction) design issues. Alternatively, you can use analytics to compare between different design concepts and approaches to find out which one creates the better user experiences (and hopefully therefore conversion rate, but keep in mind that there isn't necessarily a correlation between the two - just because users like your website doesn't mean they'll take out their wallet). There are lots of tools out there for doing analytics and user testing (Optimal Workshop is one that comes to mind, but there are plenty of others that are free/limited and will do the job without much expertise or setup).
Guerilla testing, when done in a way that doesn't focus the user's attention on the actual questions you want answered, is in essence a way to do user testing without directly involving the users. You are indirectly trying to infer information from them by some other method, and this could be through conversations or observation. What I like about guerilla testing is that the less formal and conversational you make it, the easier it is to get the information you want from the user. However, it does require that you are comfortable with talking with strangers (or observing them) without unsettling them in the process.
I would question or argue the value of obtaining user data that is not actually validated through a more rigorous or robust research methodology. But in your case the bottom line might just be to increase the conversion rate regardless of how it is achieved (be careful what you wish for though).