If we built your website but do not host it - no.
We do monitor our hosting servers. Every 5 minutes, we check about 20 different data points ranging from site response time, disk usage, memory usage, etc. If anything is outside our defined normal ranges, our support team is notified.
We do not check websites content. Meaning if your website contains a broken link, or someone on the team logged into an admin panel and made changes to a page and that page is appearing weird, or distorted - our monitoring tools would not know this. They are simply checking that your server is online and is responding to request.
If you are hosted on a server instance or above, OR have an enterprise support plan with us it is possible that we can configure some additional custom site checks regardless of if your website is hosted with us or not. Depending on the nature of these checks, additional fees may apply.