Should I use direct speech if the speech is non-sense? For example, someone says ‘qpwoeiru,’ then is saying ‘someone says qpwoeiru’ wrong but is saying “someone says ‘qpwoeiru’” right?
This is a matter of punctuation rather than grammar. Punctuation does not exist in the spoken language.
The convention in punctuation is that what is represented to be an exact replication of what was spoken or written is to be enclosed in quotation marks. That convention is independent of whether or not the person applying it believes the quotation to be nonsense.
Now it is possible to indicate uncertainty, but that does not obviate the requirement to indicate actual content by quotation marks.
I may have misheard because it makes no sense, but what I heard the Senator say was “Blum ax rept.”