Conversion of methyl group into carboxyl or amino functionality will require some sort of oxidation; most of the time these reactions are not selective and therefore have limited use in synthetic organic chemistry.
There are special cases when certain selectivity can be achieved, but they are invariably substrate and/or reagent specific. Because of this, it is impossible to give a general answer to this question - except "some sort of oxidation", but understandably, it will be hardly useful for you.
If its on an alkene or alkyne you can attach an alcohol group ( -OH), then oxidize it using CrO3 for example. If you are on a basic single bond you can use NBS hv to add a Bromine first then substitute it with OH.
Here is the chemical structure of Paraquat that I am going to work on. Do you think that is possible to convert those methyl groups into carboxyl or amino functional groups?
Methyl group can be converted into carboxyl (-COOH) group by oxidation. For example, the oxidizing agent permanganate converted toluene to benzoic acid.
You've asked in wrong categories, you're looking for organic chemistry, not analytical or clinical chemistry. Any organic chemist shall be able to answer you exactly how to do it.