There’s a variety of things that could cause this symptom, including an improperly installed timing belt. Some engines are considerably easier than others to install a timing belt. The hard ones are usually the kind where you have to route and align the belt over two separate camshaft pulleys, in addition to the crank pulley. Just one crank pulley and one camshaft pulley like my 4afe Corolla has is easier.
And if the job involved removing the engine mount, that has to be done in a way not to damage anything, b/c the engine has to be supported properly. And in many cases there are other mounts that must be loosened first, to allow for the necessary play for access. So that’s another potential cause, like if the exhaust system got damaged in the process.
Finally, another source could be just that some wire or sensor didn’t get re-connected. Timing belt jobs often require a lot of stuff be removed, and it all has to be reinstalled like it was. It’s easy for a diy’er to forget something in the process.
What I’d do if this happened to my Corolla, first I’d read out all the diagnostic codes and address those. There can be stored diagnostic codes with no check engine light. Next I’d remove the timing belt covers and check the belt tension and static alignment marks, then use a timing light to check all the sprockets are aligned like they should when the engine is running. While I was at it, I’d do a visual inspection on all the engine mounts and check the exhaust system for leaks and re-check all the connections to the sensors and other electrical and vacuum connectors.
Still a no-go? If all the routine maintenance is up to date, you’ll probably either need to live w/ it, or to take the car to a mechanic.