We don't actually know how abiogenesis occurred in the first place, which puts a bit of a limit on your question. It is fairly strongly suspected of having occurred over an area the size of a planet and timescales of at least hundreds of millions of years.
It also presumably occurred in a sterile environment, and the insides of a biological organism are very much not that sort of place. It would be quite challenging, for example, to consume food in such a way that it was totally sterilized in a hermetically sealed chamber, preventing any external microorganisms or spores or the like from entering. Preconstructed biota are likely to consume or outcompete any nascent new organisms.
The "abio-" bit is also kinda stretched when the insides are made of and filled with complex biological chemicals... sugars, proteins, fats, lots of things that simply would exist in an environment where abiogenesis is actually expected to have occurred.
The residence time of stuff inside a digestive tract is also not likely to be long enough for anything really interesting to happen anyway. Protobiological forms, like lipid bilayers, if not consumed by already-existing biota would be flushed out into the wider world.
So the answer is probably "no", unless you're talking about something world-sized with a sterilizing esophagus that has a digestive process that takes hundreds of millions of years for stuff to get from end to end.