Even though in spoken French you will hear puis-j', you always have to write "puis-je". Why? Because.

In fact, the E is never pronounced, even when the following word begins with a consonant.

The hyphen means that puis-je is pronounced as a single word - like puisje or puije. The normal rule in French is that an E at the end of a word is not pronounced, so there is no need to use an apostrophe to indicate this. Words like je, le, me, etc. are exceptions, because the E is the only vowel in the word, so the pronunciation and spelling do change before a vowel.

There are noticeable regional differences in French pronunciation - and you will hear terminal ‘e’ pronounced widely in the southern parts. Additionally, you may hear it in formal poetry recitation, even in regions where it is not pronounced in regular speech. 

https://www.talkinfrench.com/french-regional-accents/

Thanks, Maarten. Actually I did know that, but I don't think it affects my explanation of why the apostrophe is not required.