Stratified sampling is one technique of random sampling, therefore, the population must be finite and need a sampling frame a clear and complete. Meanwhile, quota sampling is non-random sampling, so that we can determine the quota of infinite population
If you consider the strict guideline on sampling technique, to perform stratified sampling you need a complete sampling frame. So, some people say to perform stratified sampling for unknown population size is not possible which I don't agree. However, getting a complete sampling frame in practical or real situation is normally very difficult to obtain. Thus, it goes back to your objective in getting the respective sample. If your study really need that each subpopulation in your study has a representative towards your sample selected, then you should go for stratified sampling, and perform systematic sampling within the strata to select your sample.
Quota sampling is a non-random sampling. So, having a complete sampling frame is not required to perform the sample selection. Thus, in my opinion knowing the exact population size is not needed to perform this technique.
Stratified sampling is one technique of random sampling, therefore, the population must be finite and need a sampling frame a clear and complete. Meanwhile, quota sampling is non-random sampling, so that we can determine the quota of infinite population