When talking about physical objects, it would be unusual to talk about injecting one thing into an object that is in the same phase - solid, liquid or gas. Not necessarily incorrect in terms of definitions, but it does not match natural usage. It's also usually only used for fluids - that is, gasses and liquids. Some solids that have partial characteristics of liquids might also be injected (powders, for instance).
In fact, injecting anything into a liquid is unusual, though as in the definition you may hear about injecting a gas into a liquid.
Often, rather than injecting the fluid into a material, it is injected into a container. For example, injecting cream into a cream cake, jam into a doughnut, or medicine into a vein. Where one of these is common, it might mutate so that you are injecting a liquid into another liquid because it is in that container. So, I would say that it is most proper to say that you inject medicine into a vein, but people will say that it is injected into the blood.
Saying that you added something to a syrup seems to me to be a very unnatural place to use the verb to inject. Instead, I would use to mix, or if another emphasis is needed, to conceal.
(Also, 'inside' is an odd choice of preposition. I would use that to talk about where the act of injecting took place, usually, but even then it's odd. It is used sometimes with inject in the way you are using, but generally to emphasise where the injected substance ends up - especially when it is concealed in a solid object.)
So, depending on the emphasis or the impression you want to give, I would use:
I mixed medicine into the syrup
I concealed medicine in the syrup
I added medicine to the syrup
The different prepositions are used with the different verbs. I don't think you could use any of the others with concealed, just in, but mixed could work with in instead of into, and added with into instead of to. The ones above are those I consider most natural.
One final note: injected will usually make people think of it being done with a syringe, if that makes sense in the specific case. As such, some people use the word specifically to suggest the use of a syringe. However, that can't make the word work in all situations. If you were using a syringe, you can indicate that explicitly:
I added medicine to the syrup with a syringe